The HectoMAP Cluster Survey. I. redMaPPer Clusters

HectoMAP is a dense redshift survey with a median redshift, z = 0.39. The average number density of galaxies with spectroscopic redshifts is ∼2000 deg⁻² (∼1200 galaxies deg⁻² are in the highly complete red-selected subsample; Geller et al. 2011; Geller & Hwang 2015; Hwang et al. 2016; Sohn et al. 2018). HectoMAP covers 52.97 deg² within the boundaries 200 < R.A. (deg) < 250 and 42.5 < decl. (deg) < 44.0. The SDSS Data Release 9 (DR9) photometric catalog (Ahn et al. 2012) is the photometric basis for the survey. The primary survey targets are red galaxies with (g − r)_model,0 > 1.0, (r − i)_model,0 > 0.5, r_petro,0 < 21.3, and r_fiber,0 < 22.0 galaxies. The color selection efficiently filters out galaxies with z ≲ 0.2, where the SDSS Main Galaxy Sample is reasonably dense. The r_fiber,0 selection removes low surface brightness galaxies that are beyond the limit of our spectroscopy.

We measured redshifts with the 300-fiber spectrograph Hectospec mounted on the MMT 6.5 m telescope (Fabricant et al. 1998, 2005) from 2009 to 2016. The Hectospec yields ∼250 spectra within a ∼1 deg² field of view in a single observation. We used the 270 mm⁻¹ grating, yielding a wavelength coverage of 3700–9150 Å with a resolution of 6.2 Å. The typical exposure time for an observation is 0.75–1.5 hr; each observation is composed of three subexposures for cosmic-ray removal. We used the HSRED v2.0 package originally developed by R.Cool and revised by the SAO Telescope Data Center (TDC) staff to reduce the data. We measured the redshifts by applying the cross-correlating code RVSAO (Kurtz & Mink 1998). Based on visual inspection of each spectrum, we classified redshifts into three categories: high quality spectra (Q), ambiguous fits (?), and poor fits (X). We use only redshifts with "Q" for scientific analyses. We obtained 58,211 redshifts for red galaxies with r_petro,0 < 21.3 and a total of 97,929 redshifts (with no color selection) in the HectoMAP region. The internal redshift error normalized by (1 + z) is ∼32 km s⁻¹.

HectoMAP is remarkably complete within the red galaxy selection limits: the survey is 98% complete to r_petro,0 < 20.5 and it is 89% complete to r_petro,0 < 21.3. Figure 1 shows the two-dimensional completeness map for HectoMAP red galaxies. The coverage is uniform over the entire survey region. These objects are the main galaxies that enter into the evaluation of the redMaPPer algorithm for cluster identification.

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Outside the red color selection, the survey completeness is patchy. We use the bluer galaxies to maximize the number of galaxies that are potential redMaPPer cluster members. Below z ∼ 0.2 the SDSS Main Sample is the primary redshift source of potential redMaPPer cluster members. The SDSS is also uniform over the HectoMAP region and the average completeness, regardless of color, is ≳90%.

Public Subaru/Hyper Suprime Cam (HSC) imaging covers ∼7 deg² of the HectoMAP region. Eventually, the entire region will be covered (Aihara et al. 2017). We use the public images in Section 3 as a partial test of the redMaPPer algorithm. In Section 3.2, we highlight the properties of the 15 redMaPPer cluster candidates imaged with Subaru.

Many studies identify galaxy cluster candidates based on photometric measures, including the red-sequence, the over-density based on photometric redshifts, and the identification of red objects associated with weak lensing peaks (Koester et al. 2007; Wen et al. 2009, 2012; Hao et al. 2010; Szabo et al. 2011; Oguri 2014; Rykoff et al. 2014, 2016; Durret et al. 2015; Oguri et al. 2018). The SDSS plays an important role in these photometric cluster surveys thanks to its wide sky coverage. Several previous catalogs based on the SDSS include cluster candidates within the HectoMAP field.

We first summarize the numbers of previous photometrically identified clusters within the HectoMAP region. Table 1 lists the number of cluster candidates in HectoMAP from MaxBCG (Koester et al. 2007), GMBCG (Hao et al. 2010), AMF (Szabo et al. 2011), WHL (Wen et al. 2009, 2012), CAMIRA (Oguri 2014), and redMaPPer (Rykoff et al. 2014, 2016). For WHL and redMaPPer (v6.3), we use the most recent versions from Wen et al. (2012) and Rykoff et al. (2016), respectively.

The number of cluster candidates in the HectoMAP region varies from 104 to 544. The number of cluster candidates depends in part on the cluster identification algorithm, the limiting survey redshift, and the richness range of candidate clusters. Table 1 shows that cluster surveys covering wider redshift ranges tend to identify more cluster candidates as expected. Within a fixed redshift range, a cluster catalog including low richness candidates contains a larger number of candidate systems. However, we do not compare the richness distributions of the various catalogs, because the definitions of richness vary substantially.

As a test of photometric catalogs, we focus on redMaPPer, a catalog that has already been compared with X-ray observations (Rozo & Rykoff 2014), SZ observations (Rozo & Rykoff 2014; Rozo et al. 2015a), and weak lensing (Simet et al. 2017). The catalog has also been compared with the SDSS and GAMA (Driver et al. 2009) spectroscopic surveys (Rozo et al. 2015b). In contrast with HectoMAP, GAMA has a median redshift of ∼0.2 (Hopkins et al. 2013). K. Rines et al. (2017, in preparation) have made a detailed test of the redMaPPer algorithm for a set on nearby rich clusters with redshifts 0.08 < z < 0.25.

HectoMAP allows for the extension of the tests of redMaPPer to the catalog limit, z ∼ 0.6. Conveniently, the magnitude limit of HectoMAP (r_petro,0 = 21.3) at z = 0.5 corresponds to M_r = − 20.96 comparable with the L_* of massive clusters (Sohn et al. 2017). Thus HectoMAP contains only redshifts of the brightest few cluster members for candidates at z > 0.5. However, even at z > 0.5 HectoMAP provides a test of the redMaPPer membership probability assignments.

As a first view of the relationship between the redMaPPer cluster sample and HectoMAP, the upper panel of Figure 2 compares the redshift distribution of HectoMAP galaxies with the distribution of redMaPPer cluster photometric redshifts. The peaks of the two redshift distributions are not coincident. For example, HectoMAP has its maximum at z ∼ 0.28, but the corresponding redMaPPer peak is at lower redshift. HectoMAP red galaxies are abundant at higher redshift, z > 0.4, where redMaPPer identifies few candidates. The lower panel of Figure 2 shows that the difference diminishes when we plot the distribution of spectroscopic redshifts of redMaPPer clusters (determined in Section 3).

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Fifteen systems lie within the the HSC Subaru Strategic Program DR1 (Aihara et al. 2017), which covers 7 deg² (∼13%) of the HectoMAP region. Figure 3 shows the HSC images of the individual systems. Each thumbnail image displays a 2' × 2' field of view around the redMaPPer center, except for HMRM08268 at z = 0.5133. Because HMRM08268 is at the edge of the public archive, we show only a 15 × 15 field of view for this cluster.

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The HSC images demonstrate that most HectoMAP-red clusters are obvious rich clusters. In a few cases, including the systems (HMRM13503, HMRM32708, HMRM34710), there are only a few red objects in the image possibly suggesting that the system is a poor group rather than a rich cluster. We note that HMRM13503 at z_phot = 0.1959 and HMRM32708 at z_phot = 0.4122 have very low redMaPPer richness λ_rich ∼ 23. For HMRM34710 at z_phot = 0.4933, the richness (λ_rich = 38.05) appears to be overestimated based both on the Subaru/HSC image and on the HectoMAP spectroscopy.

The imaging suggests that 3/15 of these redMaPPer candidates may not be rich clusters. One would expect that in these deep images the faint cluster population, invisible to the SDSS limiting magnitude, would be apparent. We comment further on the spectroscopy of these systems in Section 3.2.

As a first step in evaluating the redMaPPer candidate clusters, we measure the spectroscopic completeness for individual HectoMAP-red clusters as a function of redshift and richness (Figure 4). We define the spectroscopic completeness as

$$\begin{eqnarray}&&{f}_{\mathrm{comp}}=\displaystyle \frac{{N}_{\mathrm{RM},\mathrm{spec}}}{{N}_{\mathrm{RM}}},\end{eqnarray} \tag{ 1 }$$

where N_RM,spec is the number of redMaPPer member candidates (redMaPPer membership probability P_mem > 0) with spectra, and N_RM is the total number of redMaPPer member candidates brighter than r_petro,0 = 21.3, the limiting apparent magnitude of HectoMAP.

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For ∼90% of the clusters, HectoMAP includes redshifts for ≳50% of the member candidates. There are no strong trends of spectroscopic completeness with redshift or richness. This independence results from the r = 21.3 limit of HectoMAP that is reasonably close to the limiting r ∼ 22 used by redMaPPer. Thus, measurement of the spectroscopic properties of the HectoMAP-red clusters should be insensitive to any HectoMAP sampling biases.

With the HectoMAP sample for each HectoMAP-red cluster, we identify spectroscopic members and revise the cluster mean redshift. We identify cluster members in the phase space defined by the rest-frame relative velocity difference as a function of clustercentric distance, the R–v diagram. In the R–v diagram, the cluster members show a strong concentration around the cluster center (e.g., Diaferio & Geller 1997; Rines & Diaferio 2006; Rines et al. 2013, 2016; Serra & Diaferio 2013; Sohn et al. 2018).

Following previous studies, we identify cluster members based on the R–v diagrams. Here we apply a simple boundary because many systems are not very well populated: R_cl < 1.5 Mpc and $| {\rm{\Delta }}c({z}_{\mathrm{galaxy}}-{z}_{\mathrm{cl}})/(1+{z}_{\mathrm{cl}})| \lt 2000\,\mathrm{km}\,{{\rm{s}}}^{-1}$, where R_cl is the clustercentric distance, z_galaxy is the spectroscopic redshift of galaxies in the field, z_cl is the cluster central redshift. We set the R_cl limit based on the maximum R_cl of redMaPPer members with P_mem > 0 and the $| {\rm{\Delta }}c({z}_{\mathrm{galaxy}}-{z}_{\mathrm{cl}})/(1+{z}_{\mathrm{cl}})|$ limit based on the maximum range of spectroscopically identified members in known massive clusters (e.g., HeCS, Rines et al. 2013, 2016). The redMaPPer spectroscopic membership is insensitive to the redshift cut from ∼1500–2500 km s⁻¹. This cut is 60% or less of the photometric redshift window. The spatial and redshift limits are necessarily generous compared with techniques applicable to better sampled systems (e.g., Rines & Diaferio 2006; Rines et al. 2016; Sohn et al. 2018).

To identify cluster members, we examine the R–v diagrams based on the cluster center from the original redMaPPer cluster catalog (Rykoff et al. 2016). For the cluster central redshift, we first check the redshift of the central galaxy identified by redMaPPer, z_central. The HectoMAP redshift survey includes spectroscopic redshifts of 100 (96%) of the HectoMAP-red central galaxies. If there is a redshift for the central galaxy, we identify cluster members by applying the R_cl and $| {\rm{\Delta }}c({z}_{\mathrm{galaxy}}-{z}_{\mathrm{central}})/(1+{z}_{\mathrm{central}})|$ window. For some clusters, including the four systems without a spectroscopic redshift of the central galaxy, there are still only a few spectroscopically confirmed redMaPPer members around the central galaxy.

We estimate the median spectroscopic redshift, z_med, of the redMaPPer members with P_mem > 0. We reidentify the spectroscopic members from the R–v diagrams centered on this revised z_med. Finally, we take either z_central or z_med as the estimate of the cluster mean. We choose the estimate based on the largest number of plausible spectroscopic members. Hereafter, the cluster redshift (z_spec,cl) is the central redshift of a HectoMAP-red cluster determined from spectroscopically identified members.

Figure 5 compares the spectroscopic (z_spec,cl) and photometric redshifts (z_phot,cl) of all of the HectoMAP-red clusters. The photometric redshifts are generally consistent with the spectroscopic redshifts with a few significant offsets. The mean difference ($| {\rm{\Delta }}c({z}_{\mathrm{phot},\mathrm{cl}}-{z}_{\mathrm{spec},\mathrm{cl}})/(1+{z}_{\mathrm{spec},\mathrm{cl}})|$) is ∼3800 km s⁻¹, comparable with the mean cluster photometric redshift uncertainty for a single cluster in the redMaPPer catalog (i.e., ∼3800 km s⁻¹). It is noteworthy that the 3σ photometric error is comparable with the diameter of smaller voids in the HectoMAP survey (see Section 4.3). For the 11 systems with $| {\rm{\Delta }}c({z}_{\mathrm{phot},\mathrm{cl}}\,-{z}_{\mathrm{spec},\mathrm{cl}})/(1+{z}_{\mathrm{spec},\mathrm{cl}})| \gt 6000\,\mathrm{km}\,{{\rm{s}}}^{-1}$, our spectroscopic survey is relatively incomplete (<50%). The redMaPPer catalog identified ∼48–75 members for these 11 systems, but we identify only ∼6–35 spectroscopic members.

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Figure 6 shows the R–v diagrams of the 15 HectoMAP-red clusters with HSC images. We use the redMaPPer center and the spectroscopically determined cluster redshift.

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Figure 6 includes the R–v diagrams for the three systems that are not apparent rich clusters in the HSC images. HMRM13503 (z_phot = 0.1959) contains only eight spectroscopic members within the membership window. The redMaPPer algorithm identifies more members in the field, but these redMaPPer members are foreground and background objects. HMRM32708 (z_phot = 0.4122) has 34 spectroscopic members, but the redMaPPer richness is quite low. There are 14 spectroscopic members in the HMRM34710 (z_phot = 0.4933) field. However, only six members are located around the BCG and eight members are well separated from the BCG. Thus, HMRM13503 and HMRM34710 are poor groups as both the HSC imaging and the R–v diagrams suggest.

In 76 of the 104 clusters, the spectroscopic BCG is identical to the redMaPPer central galaxy. For four systems, we lack a spectroscopic redshift for the redMaPPer central galaxy. However, in 24 systems (∼23%), the spectroscopy identifies a BCG that is not the redMaPPer central galaxy. This fraction of offset BCGs is comparable with the redMaPPer estimate of the number of probable central galaxy misidentifications (15%–20%, Rykoff et al. 2016).

In Figure 6, gray dots mark the spectroscopic targets and red filled circles indicate HectoMAP-red member candidates with P_mem > 0.5. The dashed lines indicate the photometric redshifts of the clusters assigned by redMaPPer. Note that the photometric redshifts are often significantly offset from the mean spectroscopic redshift of the HectoMAP-red member candidates (red filled circles).

redMaPPer members generally cluster around the BCG. Interestingly, some redMaPPer members are not at the cluster redshift. These redMaPPer members tend to have low membership probability (see Section 4.1). Furthermore, a significant fraction of the spectroscopically identified members are not identified by the redMaPPer algorithm. The objects redMaPPer fails to evaluate are a mix of galaxies much bluer than the red sequence and of apparent failures of the redMaPPer algorithm to identify true red cluster members.

The R–v diagrams show some of the neighboring foreground and background structures in the line-of-sight direction toward the HectoMAP-red clusters. Figure 7 shows R–v diagrams of four HectoMAP-red clusters, where there are dense structures within the photometric redshift window for individual galaxies (indicated by gray shading). In these cases, the redMaPPer member candidate list includes large numbers of these foreground and background objects (red dots). This inclusion of nearby structures artificially inflates the redMaPPer richness of these systems.

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The redMaPPer algorithm identifies clusters on the basis of the red sequence. To explore the prominence of the red sequence, Figure 8 shows the (g − r)_model,0 versus r_petro,0 color–magnitude diagrams of the 15 HectoMAP-red clusters with HSC images. We plot galaxies within 15 arcmin of each cluster center as gray dots. The red filled circles and black open circles indicate redMaPPer members and the full set of spectroscopically identified members, respectively. Following Rines et al. (2013), we determine the g − r red sequence of each cluster by assuming a slope of −0.04 in the color–magnitude domain and fitting the spectroscopically identified member galaxies; we then identify red-sequence members as objects within ±0.2 mag of the red sequence. The slope of the red sequence may change at higher redshift, but galaxies in the well-populated HectoMAP X-ray clusters (z ≲ 0.4, Sohn et al. 2018) are consistent with this definition of the red sequence.

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The red sequence we use differs from the one used by redMaPPer. We determine the red sequence in the (g − r)_model,0 − r_cmodel,0 space; redMaPPer derives the multicolor red-sequence model based on all of the SDSS bands. Here, we use the red sequence only to show the impact of spectroscopy on redMaPPer cluster identification. We do not use the red sequence itself to identify real systems. However, for 11 of the 15 clusters the g − r versus r red sequence is well-defined. When this red sequence is not readily visible, the richness of the system appears to be substantially overestimated by redMaPPer and/or the system is at z ≥ 0.4 where the scatter in the red sequence is substantial in this color–magnitude space.

We examine the (r − i)_model,0 − i_petro,0 color–magnitude diagrams for clusters with z ≥ 0.35. For the clusters at z > 0.35, we determine the r − i red sequence for each cluster by assuming a slope −0.01 and we select red-sequence members within ±0.1 of the red sequence. This choice of slope provides a reasonable representation of the data. In general, the red sequences of clusters with z ≥ 0.35 are flatter and tighter in the r − i color domain (e.g., Figure 3 from Rykoff et al. 2014). The scatter around the red sequence increases with redshift (e.g., Figure 4 from Rykoff et al. 2014).

In the color–magnitude diagram for each cluster, a significant number of nonmembers (generally background objects) contaminate the apparent red sequence and bias the richness estimate upward. There are also spectroscopically determined members that lie on the approximate red sequence we define, but they do not have a redMaPPer membership probability. This problem may originate from large offsets between the photometric redshift reported by redMaPPer and the more accurate spectroscopic mean redshift.

Spectroscopic surveys of galaxy clusters allow for the estimation of the contamination of candidate members selected by color. Figure 9 shows the spectroscopically identified member fraction among the redMaPPer candidate members. We define the spectroscopically identified member fraction in each cluster as

$$\begin{eqnarray}&&{f}_{\mathrm{spec} \mbox{-} \mathrm{mem},\mathrm{cl}}=\displaystyle \frac{{N}_{\mathrm{spec} \mbox{-} \mathrm{mem},\mathrm{RM},\mathrm{spec}}}{{N}_{\mathrm{RM},\mathrm{spec}}},\end{eqnarray} \tag{ 2 }$$

where N_{spec-mem,RM,spec} is the number of spectroscopically identified members among the redMaPPer candidate members and N_RM,spec is the total number of redMaPPer candidate members with P_mem > 0 and with spectra. The median f_spec-mem,cl of the HectoMAP-red clusters is ∼59%. The median f_spec-mem,cl increases for candidate members with P_mem > 0.5 (∼72%) and P_mem > 0.9 (∼91%). We find no dependence of the spectroscopically identified member fraction on redshift and richness.

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The R–v diagrams in Figure 10 show how the distribution of the spectroscopic HectoMAP-red cluster members depends on P_mem. We plot stacked R–v diagrams of the 61 low redshift (z ≤ 0.35) and 43 high-redshift (z > 0.35) HectoMAP-red clusters within three different P_mem bins: P_mem ≥ 0.9, 0.9 > P_mem ≥ 0.5, and 0.5 > P_mem. For P_mem ≥ 0.9, there is a strong concentration of spectroscopically identified members toward the cluster center. Even for these high confidence objects, ∼11% have a rest-frame relative velocity that differs from the cluster center by ($| {\rm{\Delta }}{cz}/(1+{z}_{\mathrm{cl}})| \geqslant 2000\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The typical velocity dispersion derived from the stacked P_mem ≥ 0.9 members is ∼650 km s⁻¹ comparable with the typical rich cluster line-of-sight velocity dispersion (∼700 km s⁻¹, Rines et al. 2013). The candidate members with lower P_mem generally lie at larger radius if they are within the redshift range for spectroscopic membership. The fraction of objects with large $| {\rm{\Delta }}({cz})/(1+{z}_{\mathrm{cl}})|$ increases as P_mem decreases: ∼35% are outliers when 0.9 > P_mem ≥ 0.5 and ∼66% outliers are when 0.5 > P_mem.

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Based on the fraction of spectroscopically identified members among redMaPPer member candidates, we provide a correction function for redMaPPer cluster membership. Rozo et al. (2015b) also estimate the spectroscopic member fraction among redMaPPer members, but they use only red galaxies from various spectroscopic samples.

Figure 11 shows the spectroscopically identified member fractions as a function of P_mem. A large fraction, but not all of the objects with P_mem > 0.9 tend to be spectroscopically identified members. We examine the spectroscopically identified member fractions at various clustercentric radii within different magnitude ranges, but the overall fractions have little dependence on these observables. The spectroscopically identified member fractions are also insensitive to the redshift and the richness of clusters.

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We fit the spectroscopically identified member fraction with a simple linear relation,

$$\begin{eqnarray}&&{f}_{\mathrm{spec} \mbox{-} \mathrm{mem}}=(0.25\pm 0.02)+(0.66\pm 0.02){P}_{\mathrm{mem}}.\end{eqnarray} \tag{ 3 }$$

This relation provides a spectroscopically determined correction to the redMaPPer membership probability. It is interesting that at the lowest redMaPPer probabilities, there are more real spectroscopic members (∼14%) than the redMaPPer algorithm would suggest.

We compare the f_spec-mem from the HectoMAP-red sample to a similar relation derived from the HeCS-red sample (K. Rines et al. 2017, in preparation). The HeCS-red sample includes 23 high-richness (λ ≥ 64) redMaPPer clusters in the redshift range 0.08 < z < 0.25. K. Rines et al. (2017, in preparation) examine the spectroscopically identified member fraction of the HeCS-red sample based on extensive SDSS and MMT/Hectospec redshift data. The blue dotted line in Figure 11 shows the linear relation for the HeCS-red sample; this relation is much shallower than for the HectoMAP-red sample. In the HeCS-red sample, fewer high P_mem galaxies are spectroscopically identified members and more low P_mem galaxies are spectroscopically identified members.

The HeCS-red clusters include the highest richness systems at low redshift; in contrast, the HectoMAP-red clusters are low richness systems covering a much wider redshift range. In its redshift range 0.08 < z < 0.25, the redshift survey for the HeCS-red has more complete (∼90%) coverage of the red sequence, but reaches only r ≤ 20.5. The r − i cut in HectoMAP leads to undersampling of the red sequence in this redshift range. In addition, K. Rines et al. (2017, in preparation) use the caustic technique (Diaferio & Geller 1997; Diaferio 1999; Serra & Diaferio 2013) for identifying spectroscopic members. This approach is more stringent than our coarse membership determination. These substantial differences in samples and member identification probably explain the differences in the spectroscopically identified member fractions.

Spectroscopy of the HectoMAP-red clusters confirms the identification of most redMaPPer systems. Of these systems, 90% or more show a concentration in the R–v diagram, and the R–v diagram identifies more than 10 members. Overall, the redMaPPer catalog has impressive purity: ≳90% of the candidate systems are condensations in redshift space. Based on the spectroscopy, we refine the cluster mean redshift and richness.

The upper panel of Figure 12 shows the number of spectroscopically identified members in the HectoMAP-red clusters as a function of redMaPPer richness, λ_rich. We next test the correlation between the number of spectroscopically identified members and the redMaPPer richness (λ_rich). The Pearson correlation coefficient (0.13 ± 0.10) is very low. If we divide the samples at z = 0.35, where redMaPPer identifies clusters based on a different color–magnitude domain, neither sample shows a significant correlation (0.49 ± 0.11 and 0.21 ± 0.15, respectively).

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The 10 HectoMAP-red clusters with fewer than 10 spectroscopic members cover a wide redMaPPer richness range 20 < λ_rich < 70. Some of these systems may be poor groups. The HSC image of one of these candidate systems, HM13503, shows that only a few (∼5) redMaPPer candidate members are luminous ellipticals. Eight high-redshift (z > 0.35) candidates with fewer than 10 spectroscopic members may be poorly sampled; many members have r > 21.3.

Based on the spectroscopy, we construct a corrected redMaPPer richness (λ_rich,cor). To compute this corrected richness, we use spectroscopically identified members that also have a redMaPPer membership probability (P_mem > 0). At redshift z > 0.3, nearly all spectroscopically identified members have a redMaPPer membership probability; at z ≲ 0.4, the median fraction is ∼70%. The global fraction of spectroscopically identified members without a redMaPPer P_mem is ∼25%.

The corrected richness is

$$\begin{eqnarray}&&{\lambda }_{\mathrm{rich},\mathrm{cor}}=\sum {f}_{\mathrm{spec} \mbox{-} \mathrm{mem}}=\sum (0.25+0.66{P}_{\mathrm{mem}}).\end{eqnarray} \tag{ 4 }$$

In Equation (3), we use only redMaPPer candidate members with r_petro,0 < 21.3, the limiting apparent magnitude of the HectoMAP survey. The corrected richness indicates the total number of redMaPPer members after correction for contamination by line-of-sight objects brighter than the HectoMAP limit. The weighting of objects without spectroscopically confirmed membership reflects the original redMaPPer prescription with the correction to the probabilities that we derive from spectroscopy. We have checked that the correction to the redMaPPer membership probability (Figure 11) is insensitive to apparent magnitude, redshift, and color.

The lower panel of Figure 12 displays the corrected richness as a function of the original redMaPPer richness (λ_rich). For the overall and high-z (z > 0.35) samples, the corrected richness is not tightly correlated with the original redMaPPer richness; the Pearson correlation coefficient is 0.57 ± 0.08. The incompleteness of our spectroscopic sample at 21.3 < r < 22, where most redMaPPer members in the high-z samples appear, may produce the lack of correlation. In contrast, the lower redshift sample shows a significant correlation with a coefficient of ∼0.99. For low-z samples, the HectoMAP redshift survey covers a significant fraction of redMaPPer members. The linear fit between λ and λ_rich,cor for the z ≤ 0.35 HectoMAP-red clusters (after 2σ clipping) is

$$\begin{eqnarray}&&{\lambda }_{\mathrm{rich},\mathrm{cor}}=(1.3\pm 7.8)+(1.2\pm 0.2)\,{\lambda }_{\mathrm{rich}}.\end{eqnarray} \tag{ 5 }$$

Figure 13 is similar to Figure 12, but for the raw redMaPPer richness (λ_rich/S). The raw richness indicates the effective number of redMaPPer members brighter than the limiting magnitude of the survey (r ∼ 22). The richness (λ_rich) is a correction of the raw richness (λ_rich/S) for the geometric survey mask (including star holes and survey boundaries) within the survey area, or the missing galaxies below the redMaPPer magnitude limit i < 21.0 (Rykoff et al. 2014, 2016; Rozo et al. 2015b).

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The raw richness shows a weak correlation with the number of spectroscopically identified members in both the low- and high-redshift samples. The Pearson correlation test yields marginal correlation coefficients: 0.48 ± 0.11 for z ≤ 0.35 and 0.65 ± 0.12 for z > 0.35. The dashed and dotted lines in Figure 13 indicate linear fits for the low- and high-redshift samples, respectively. However, the spectroscopically corrected richness is tightly correlated with the raw richness, with high correlation coefficients (0.97 ± 0.03 and 0.97 ± 0.03). We derive linear fits between the spectroscopically corrected richness and the redMaPPer raw richness (λ_rich/S) for both redshift ranges

$$\begin{eqnarray}&&{\lambda }_{\mathrm{rich},\mathrm{cor}}=(-3.1\pm 7.4)+(1.4\pm 0.2)({\lambda }_{\mathrm{rich}}/S)\,\mathrm{for}\,z\leqslant 0.35,\end{eqnarray} \tag{ 6 }$$

and

$$\begin{eqnarray}&&{\lambda }_{\mathrm{rich},\mathrm{cor}}=(7.0\pm 7.5)+(1.0\pm 0.2)({\lambda }_{\mathrm{rich}}/S)\,\mathrm{for}\,z\gt 0.35.\end{eqnarray} \tag{ 7 }$$

Several studies suggest that the redMaPPer λ_rich is a mass proxy (Rozo et al. 2015a, 2015b; Simet et al. 2017). However, HectoMAP spectroscopy suggests that λ_rich is not tightly correlated with the spectroscopically corrected richness at z > 0.35. In this redshift range, the redMaPPer raw richness correction S is significant. Extension of the spectroscopy to fainter magnitudes for a sufficiently large sample of photometrically identified clusters would substantially improve the calibration of λ_rich as a mass proxy, in particular, at z > 0.35.

Figure 14 shows the position of the HectoMAP-red clusters in the cone diagram for the HectoMAP region. We mark the positions of the HectoMAP-red clusters (red circles) based on the HectoMAP-red spectroscopic redshifts. If we plotted the HectoMAP-red clusters based on the redMaPPer photometric redshift, the systems would be offset from the galaxy overdensities along the line of sight.

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Figure 14 also shows that some of the obvious dense HectoMAP regions do not have HectoMAP-red clusters associated with them. In some of these regions, fingers in redshift space are apparent. For example, massive clusters associated with X-ray emission (cyan circles, Sohn et al. 2018) are missing from the HectoMAP-red cluster sample.

We can obtain some estimate of the completeness of the redMaPPer cluster survey by comparing it with the HectoMAP X-ray clusters (Sohn et al. 2018) that represent some of the most massive systems in the HectoMAP field with M₂₀₀ ≳ 2 × 10¹³ M_⊙. The X-ray clusters are identified by a coidentification method based on a friends-of-friends algorithm applied to HectoMAP along with X-ray source detection from the ROSAT all sky survey data. There are 15 HectoMAP X-ray clusters with 0.03 < z < 0.40. All of these clusters have at least 18 spectroscopic members; the richest of them has 218 spectroscopically confirmed members.

We identify HectoMAP-red cluster counterparts to the X-ray systems if they are within 1.5 Mpc and $| {\rm{\Delta }}{cz}/(1\,+{z}_{\mathrm{cl}})| \lt 1000\,\mathrm{km}\,{{\rm{s}}}^{-1}$. We set the 1.5 Mpc criterion to reflect the redMaPPer limiting size (Figure 6). Of the HectoMAP X-ray clusters, 11 have HectoMAP-red cluster counterparts in the public redMaPPer catalog. One HectoMAP X-ray cluster is at z = 0.03, below the redMaPPer survey limit. The three missing X-ray clusters are massive systems with large velocity dispersion (≳450 km s⁻¹, Sohn et al. 2018). The dynamical masses of two of the missing X-ray clusters are larger than the effective mass cut (≳1.24 × 10¹⁴ M_⊙, Simet et al. 2017) of the public redMaPPer catalog with λ > 20. These X-ray clusters show obvious g − r or r − i red sequences (Sohn et al. 2017 submitted). These missing X-ray clusters (20 ± 11%) are listed in a private redMaPPer catalog with a lower richness threshold λ > 5 (E. Rozo 2018, private communication). Tests of photometric catalogs against all sky X-ray data with deeper and better resolution from e-ROSITA combined with a dense redshift survey for some significant sample can provide a much more robust foundation for cosmological applications than that currently available.