SHELS: TESTING WEAK-LENSING MAPS WITH REDSHIFT SURVEYS

The construction of large catalogs of clusters of galaxies has a long history in the optical (e.g., Abell 1958; Zwicky et al. 1968; Abell et al. 1989; Koester et al. 2007) and in the X-ray (e.g., Schwartz 1978; Piccinotti et al. 1982; Edge et al. 1990; Ebeling et al. 1996; Böhringer et al. 2004). In these approaches, the catalog selection parameters are related to the cluster mass only through scaling relations. In contrast, weak lensing offers the enticing possibility of obtaining a "(projected) mass-selected" catalog of clusters directly.

Several studies show that weak-lensing maps are a route to cluster identification. Wittman et al. (2001) used the DLS to make the first detection of a previously uncataloged cluster from a convergence map. Subsequently, Miyazaki et al. (2002, 2007), Hetterscheidt et al. (2005), Wittman et al. (2006), Schirmer et al. (2007), Gavazzi & Soucail (2007), Hamana et al. (2009), Bergé et al. (2008), and Dietrich et al. (2007) have all demonstrated coincidence of optical and/or X-ray clusters with peaks in weak-lensing convergence maps.

The number of candidate clusters identified from weak-lensing maps has risen steeply to several hundred. For the two largest sets of candidates, the success rate for identifying these candidates with clusters of galaxies differs substantially. Miyazaki et al. (2007) claim an 80% success rate for the ∼100 peaks in their convergence maps with a signal to noise greater than 3.69. Hamana et al. (2009) support this claim with sparse spectroscopic sampling of 36 weak-lensing cluster candidates. In contrast, Schirmer et al. (2007) identified 158 possible mass concentrations in a blindly selected sample and derive a ∼45% success rate for their ∼4σ convergence map peaks consistent with an earlier evaluation of a subsample of the survey (Maturi et al. 2007). The 4σ significance threshold may be an overestimate because Schirmer et al (2007) combine sets of peaks identified with two different statistics to construct their sample. A third independent analysis of convergence maps from the CFHTLS (Gavazzi & Soucail 2007) yields an intermediate ∼65% success rate among 14 peaks identified above a signal-to-noise threshold of 3.5.

Miyazaki et al. (2007) attribute the much lower success rate of Schirmer et al. (2007) to the generally lower source density in the Schirmer et al. (2007) weak-lensing maps. However, there are other possibly important differences in the construction and analyses of the maps. Schirmer et al. (2007) include a detailed discussion of a variety of statistical issues in both the identification of convergence map peaks and of the related concentrations of galaxies. Although there are some qualitative aspects in their evaluation of the galaxy counts, they limit the identification of coincidences to apparent systems with redshifts less than ∼0.3. This limit is roughly consistent with their lower source density. In contrast, Miyazaki et al. (2007) and Hamana et al. (2009) claim probable coincidences to a redshift of 0.5 or more. Short of performing exactly the same analysis on the two data sets, it is difficult to account for the differing success rates from consideration of the relative background source counts alone.

Miyazaki et al. (2007) use a combination of X-ray observations, redshift measurements (extended by Hamana et al. 2009), and imaging to identify their convergence peaks with systems of galaxies. Schirmer et al. (2007) take a more uniform approach of counting galaxies in beams with a 2' radius around the centers of their convergence peaks; they measure the apparent overdensity in the magnitude range R ∼ 17–22 and make an assessment of significance depending on the excess count (with some occasional qualitative modifications).

Gavazzi & Soucail (2007) also take a uniform approach to the evaluation of significance of the peaks; they use the distribution of photometric redshifts for galaxies in a circular aperture of 2' centered on each convergence peak. They subtract a background distribution computed from galaxies which lie more than 6' from any convergence peak. For nine of their 14 peaks, they define the associated cluster redshift as the location of the most prominent photoz peak. One remarkable feature of the Gavazzi & Soucail analysis is the low velocity dispersion (as low as 450 km s⁻¹) for some of the systems corresponding to convergence peaks.

Most recently, Kubo et al. (2009) use X-ray observations, DLS images, and available redshifts to estimate the fraction of false positives in a maximum likelihood weak-lensing reconstruction. They conclude that only 10%–25% of their peaks with signal to noise greater than 3.5 are false positives. This conclusion may be overly optimistic because they consider coincidences of peaks and apparent systems with angular separations as large as 4 arcmin. Schirmer et al. (2007) and Gavazzi & Soucail (2007) show that the average angular offset between convergence peaks and optical counterparts is 09 ± 05. The convergence maps of Schirmer et al. (2007) and Gavazzi & Soucail (2007) are sensitive to lensing systems in approximately the same redshift range as the DLS.

Recent simulations of the efficacy of convergence maps for construction of mass-selected cluster catalogs are based on ray-tracing through large n-body simulations (Hamana et al. 2004; Hennawi & Spergel 2005). Hamana et al. (2004) make a set of mock observations where they associate convergence map peaks with halos. They conclude that for convergence peaks with a signal to noise > 4, more than 60% of massive halos are detected. On the other hand, they also estimate that a conservative 40% of these convergence peaks are false positives. Although the details of the simulation differ from the observations, the estimated false positive rate is consistent with the conclusions of Gavazzi & Soucail (2007), more optimistic than Schirmer et al. (2007), and less optimistic than Miyazaki et al. (2007) or Kubo et al. (2009). The ∼80% rate of the Miyazaki et al (2007) survey is very close to the ∼85% maximum achievable (Hennawi & Spergel). Simulations by Dietrich et al. (2007) also show that 75% of matches between convergence map peaks and massive halos are within 215, consistent with the observational offsets observed by Schirmer et al. (2007) and Gavazzi & Soucail (2007).

In all of these recent studies, some convergence peaks are associated with superpositions of several systems, often with low velocity dispersion, along the line of sight toward the weak-lensing peak. A clean evaluation of the nature and frequency of such superpositions is very difficult without a complete foreground redshift survey. The fraction of "dark" peaks attributable to noise appears to vary with the nature of the convergence map and with the method of cross-identifying convergence peaks with physical systems. Yet another issue not considered in analyses to date is the frequency of accidental coincidences between convergence map peaks and apparent systems of galaxies. Understanding the noise properties of convergence maps and the related correspondence between the weak-lensing signature and structures in the galaxy distribution is a continuing challenge (Mellier 1999; Schneider 2006).

A foreground redshift survey is an independent measure of the matter distribution revealed by a convergence map. Here, we use a complete redshift survey, SHELS, to compare peaks in the DLS convergence map with probable systems identified in the redshift survey. We develop procedures similar to those adopted by Schirmer et al. (2007) and Gavazzi & Soucail (2007) who use narrow probes through their photometric data to evaluate coincidences between potential lensing systems and peaks in their convergence maps.

The advantages of a redshift survey for evaluating the efficiency of a lensing include (1) resolution of structures along the line of sight, (2) availability of a velocity dispersion, and (3) bases for constructing a control sample and evaluating false positives. Combining a redshift survey with a lensing map enables an evaluation of the frequency of chance coincidences between significant lensing peaks and systems of galaxies or halos. The redshift survey also provides an estimate of the number of systems undetected by the weak-lensing map. Consideration of these issues has previously been limited largely to simulations.

In Section 2, we describe the DLS lensing map and the SHELS redshift survey. We calculate the sensitivity of the DLS to clusters with a given rest-frame line-of-sight velocity dispersion in Section 2.2. Section 3.1 develops the technique we use for evaluating the match between the convergence map peaks and structures in the redshift survey. We examine the redshift distribution along the line of sight toward the 12 most significant weak-lensing peaks in Section 3.2 and compute rest-frame line-of-sight velocity dispersions for these candidate DLS/SHELS clusters in Section 3.3. We discuss the seven known extended clusters X-ray sources in Section 3.4 and construct a set of SHELS candidate clusters independent of the DLS map in Section 3.5. We evaluate the efficiency and completeness of the DLS convergence map in Section 4. In Section 5, we demonstrate the potential impact of photometric redshifts on the evaluation of the efficiency of weak lensing for massive cluster identification. Section 6 compares our results with simulations by Dietrich et al. (2007) and Hamana et al. (2004). We conclude in Section 7. We use the WMAP concordance cosmology (Spergel et al. 2007) throughout with H_o = 73, Ω_m = 0.3, and Ω_Λ = 0.7.

Photometric observations of F2 were made with the MOSAIC-I imager (Muller et al. 1998) on the KPNO Mayall 4 m telescope between 1999 November and 2004 November. The R-band exposures, all taken in seeing <09 FWHM, are the basis for the weak-lensing map of F2. The total exposure is 18,000 s; the 1σ surface brightness limit in R is 28.7 mag arcsec⁻² yielding about 45 resolved sources per square arcminute. Wittman et al. (2006) describe the reduction pipeline.

Kubo et al. (2009) carry out a maximum likelihood lensing reconstruction of F2. They base the convergence (κ) map on sources in the range 22.0 < R < 25.5. After removal of objects which are too small relative to the point-spread function (PSF), so large that they are probably nearby, or so elongated that they are probably superpositions (Wittman et al. 2006), the source catalog contains 328,000 galaxies or 23 galaxies arcmin⁻². This source density is within the range of other ground-based surveys (Miyazaki et al. 2007; Schirmer et al. 2007; Gavazzi & Soucail 2007). We use this map for our evaluation of the efficacy of the Kubo et al. (2009) weak-lensing maps for locating massive clusters.

Kubo et al. (2009) construct noise maps of F2 with 100 realizations based on the source positions in the original catalog. In each realization, we assign ellipticities randomly to sources. We define σ_DLS as the rms deviation in the noise maps. We define the signal to noise as ν = κ/σ_DLS. At each point in the image, we produce a ν map by dividing the reconstructed κ value by the rms variation in the random realizations at the same spatial positions. Kubo et al. (2009) select candidate shear peaks as local maxima in this ν map which has a pixel scale of 1.5 arcmin pixel⁻¹.

In the F2 field of the DLS, Kubo et al. (2009) identify 12 peaks with ν≳ 3.5. Although Kubo et al. (2009) construct their list of potential peaks using SExtractor with a 9-pixel area filter, the peak significance for selecting the ν>3.5 peaks is determined by the signal to noise in the highest significance pixel. Kubo et al. (2009) also estimate a total signal to noise for their most significant peaks. However, their method cannot resolve overlapping peaks like those corresponding to the two massive clusters comprising Abell 781 (see Geller et al. 2005). Thus, we do not use the Kubo et al. (2009) total signal-to-noise values here.

Kubo et al. (2009) tentatively identify 10 of these weak-lensing peaks with plausible systems of galaxies. These cross-identifications are based on inhomogeneous data in an approach similar to Miyazaki et al. (2007) and the success rate is similar. One concern in some of these identifications (also noted by Kubo et al. 2009) is that the separation between the center of the convergence peak and the galaxy system is as large as 4'. Schirmer et al. (2007) and Gavazzi & Soucail (2007) find that the typical separation of convergence peaks and plausibly associated galaxy systems is about an arcminute.

To estimate the completeness of the weak-lensing-selected cluster catalog derived from the κ map (Kubo et al. 2009), we calculate the sensitivity limits as a function of redshift. For any individual background galaxy in the weak-lensing limit, the induced measured tangential ellipticity (and hence the signal-to-noise ratio (S/N) for detection) depends on both the angular diameter distance to the galaxy and on the size of the galaxy relative to the PSF. For a collection of galaxies, the effective total weight as a function of redshift is a polarizability weighted sum over the individual distance ratios. The effective distance ratio for the ensemble of background galaxies is

$$\begin{equation} W_{\rm eff}(z_l) = {{\sum _{s} \frac{D_A(0,\,z_l) D_A(z_l,\,z_s)}{D_A(0,\,z_s)}} W_s \over {\sum _{s} W_s }}, \end{equation} \tag{ 1 }$$

where the subscripts s and l refer to the sources and lens, respectively. W_s is the weight of each source galaxy (which depends on the object size), z_l is the lens redshift, D_A(z₁, z₂) is the angular diameter distance between z₁ and z₂, (the ratio is zero if z_s < z_l), and the sum is over all sources. For a realistic survey, an additional suppression results from misidentification of faint foreground galaxies, each of which should have weight zero. Furthermore, in a ground-based survey, a large number of faint galaxies are unresolved. Kubo et al. (2009) reject objects with FWHM smaller than 1.2 times the PSF size; thus W_s = 0 for these objects.

For each redshift, we compute the angular diameter distance factors in Equation (1) for each redshift assuming the "concordance" cosmology (Spergel et al. 2007). We derive the W_s terms using data from the COSMOS Subaru galaxy catalog (Taniguchi et al. 2007) and the COSMOS ACS catalog (Leauthaud et al. 2007). The COSMOS combined ACS/Subaru data set is currently the best data set for this calibration because it covers by far the largest area with both ground- and space-based resolution to a depth comparable to the DLS. In addition, the Subaru multi-color observations provide photometric redshift estimates (Mobasher et al. 2007) for the majority of galaxies in the survey area. We use the COSMOS catalog size and photometric redshift information to estimate the redshift distribution of the galaxies that would be resolved in DLS. The comparison of ground-based and space-based COSMOS images also provide a measure of how well-resolved DLS galaxies are as a function of their DLS size. Thus, the COSMOS data provide a route to a proper weighting of the DLS galaxies as a function of their probable intrinsic size in Equation (1). We match galaxies in the COSMOS Subaru and Advanced Camera for Surveys (ACS) catalogs with the additional requirement that the catalog magnitudes in I match to within 0.3 mag. (This restriction eliminates errors in matching the two catalogs; it also eliminates most objects that cannot be adequately separated from neighboring objects in the Subaru imaging.)

To construct a galaxy sample equivalent to the DLS sample, we select all galaxies from the COSMOS galaxy catalog with Subaru r magnitude in 20 < r < 25.3 (a rough match to the DLS R magnitude), scale radius >03 (to ensure that the measured size when convolved with the DLS PSF is larger than the size cutoff in the DLS catalogs), and 0.01 < z_phot < 2.5. The last cut removes some potentially catastrophic redshift errors; we choose the high-redshift cutoff ≫z_max, the maximum lens redshift considered, to avoid biasing the high-redshift tail of the sensitivity. The cut on size produces an effective mean source redshift in the range 0.7–0.8. MMT spectroscopy of a small subset of the DLS sources is consistent with this estimate (Geller et al. 2005). Approximately 29,000 COSMOS objects meet all of the criteria yielding a source density comparable (but about 15% lower) with the source density of objects in the weak-lensing reconstruction of DLS field F2. For each object, we compute the FWHM convolved with the mean PSF for the DLS data, and we compute the weight for that size (and PSF) as in Wittman et al. (2006). For each lens redshift, we compute W_eff.

Because we identify clusters from the weak-lensing map based on the projected surface density within a single reconstruction pixel (15 × 15), the S/N should scale linearly with the mass enclosed within that surface area. Thus, we normalize the redshift dependence of our sensitivity by comparing to simulations. To calculate the absolute sensitivity of the weak-lensing survey to massive halos, we follow a modified version of the procedure for generating simulated catalogs in Khiabanian & Dell'Antonio (2008). We generate simulated galaxies with the same size–magnitude and magnitude–redshift relations as the Hubble Deep Field (HDF) North and South fields (Williams et al. 1996; Casertano et al. 2000) using the prescription of Khiabanian & Dell'Antonio (2008). We use these simulated galaxies to populate seven logarithmically spaced redshift shells. We then distort the images with a lens (modeled as a core-softened cutoff isothermal sphere with a given rest-frame line-of-sight velocity dispersion, σ_iso, a 1'' core radius, and a 100'' cutoff radius). We resample the distorted images with the MOSAIC pixel scale, convolved with the model DLS PSF, and add noise to match the noise in the F2 image. We repeat this procedure for different values of σ_iso (at a fixed z_lens = 0.3) to generate maps of the weak-lensing signal. We run the Kubo et al. (2009) detection method on these images to determine the value of σ_iso where the peak S/N reaches the ν = 3.5 cluster selection limit in Kubo et al. (2009). In Figure 1, we plot the detection limit as a function of redshift (L(z) = L(z = 0.3)W_eff(z = 0.3)/W_eff(z)); the solid line represents ν = 3.5 and the dotted lines represent the detection limits for ν = 3 and ν = 4, respectively. Because the detection criteria for the simulated clusters match the Kubo et al. (2009) sample, the simulation may overpredict the signal from clusters at very low redshift (z < 0.1); the angular region where we calculate the potential in the reconstruction encloses significantly less mass (or alternatively, we spread the same signal over multiple pixels).

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There are some fundamental limitations on the accuracy of our survey sensitivity calculations. First, and most important, there is cosmic variance in the number of background galaxies behind a cluster. For clusters near the detection limit, the shear signal is detectable in a region containing only 1000–2000 galaxies; on those scales there are myriad effects that can greatly alter the number of background galaxies. Second, although the COSMOS/Subaru field is extremely valuable, it covers ≲1 deg² of the sky. Therefore, any systematic differences between the properties of the COSMOS field and DLS F2 can change the normalization. One of the results of the analysis of the different DLS fields is that there are strong field-to-field variations in the types and amounts of structure present on degree scales. Without large surveys, there is no way around this fundamental limitation.

There are also several more accessible sources of uncertainty in the sensitivity calibration technique. First, the computed mass sensitivity depends somewhat on the model mass profile; more centrally concentrated clusters are more easily detected. Second, because we select galaxies in the COSMOS/Subaru r band, which is not identical to the DLS R band, there is a slight bias in the redshift and size distributions of the COSMOS galaxies. Finally, the photometric redshift estimates introduce a bias if the faint galaxies have a significantly different redshift distribution from the brighter ones. However, all these sources of uncertainty change the calibration by less than ∼10%. For example, taking the magnitude cutoff in the COSMOS catalog as 24.8 instead of 25.3, a much larger change than the uncertainty in the magnitude calibration, changes the redshift of peak sensitivity by 0.01, and changes the sensitivity at z = 0.4 by 4%. In the analysis of the incompleteness of weak-lensing cluster detection, we take a 10% uncertainty in the sensitivity limit into account when determining which clusters selected from the redshift survey should be detected in the lensing map.

We investigate the association between convergence peaks and clusters (halos) in the galaxy distribution based on a complete redshift survey. Our goal is identification of the systems of galaxies that should produce a weak-lensing signal based on Figure 1. We thus identify systems in the redshift survey with rest-frame line-of-sight velocity dispersion ≳500 km s⁻¹ (see Section 3).

We constructed the galaxy catalog from the R-band source list for the F2 field of the DLS. We used surface brightness to separate stars from galaxies (see M. J. Kurtz et al. 2010, in preparation for details). The final catalog contains 9825 galaxies with R ⩽ 20.3; 9603 of these galaxies have redshifts.

We acquired spectra for the objects with the Hectospec (Fabricant et al. 1998, 2005) on the MMT from 2004 April 13 to 2007 April 20. The Hectospec observation planning software (Roll et al. 1998) enables efficient acquisition of a magnitude limited sample.

The SHELS spectra cover the wavelength range 3500–10000 Å with a resolution of ∼6 Å. Exposure times ranged from 0.75 to 2 hr. The lowest surface brightness objects in the survey required the longer integrations. We reduced the data with the standard Hectospec pipeline (Mink et al. 2007) and derived redshifts with RVSAO (Kurtz & Mink 1998) with templates constructed for this purpose (Fabricant et al. 2005). Our 1151 galaxies with repeat observations yield robust estimates of the median error in cz where z is the redshift. For emission line objects, the median error (normalized by (1 + z)) is 27 km s⁻¹; the median for absorption line objects (again normalized by (1 + z)) is 37 km s⁻¹ (see also Fabricant et al. 2005).

The integral completeness of SHELS to R = 20.3 is 97.8%; the differential completeness at the limiting magnitude is 94.6%. The 218 objects (out of 9825) without redshifts are low surface brightness blue objects or objects near the survey corners and edges. Figure 2 shows the completeness of SHELS as a function of apparent magnitude (note that the completeness scale ranges from 0.9 to 1.0).

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The SHELS survey also includes 1871 galaxies with 20.3 < R ⩽ 20.6; the survey is ∼60% complete in this magnitude interval. The completeness is patchy across the field but is generally at or above the mean in the region of the most significant weak-lensing peaks.

The median redshift of SHELS is 0.295 for the magnitude limited sample with R ⩽ 20.3. Figure 3 shows the redshift distribution for the survey to a limiting R = 20.3. The impact of large-scale structure is obvious to a redshift z ∼ 0.55.

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Schirmer et al. (2007) and Gavazzi & Soucail (2007) develop uniform procedures for evaluating the coincidence of concentrations of galaxies with convergence map peaks. Schirmer et al. (2007) compare the galaxy counts with R ∼ 17–22 in a cone of 2' radius centered on a lensing peak with the galaxy density in the surrounding field. They assign significance classes to the excess counts based on Poisson statistics. Gavazzi & Soucail use the distribution of photometric redshifts for galaxies brighter than i' = 23 in 2' radius probes. They subtract a background photometric redshift distribution from each of the probes. They then compute the background from the survey area more than 6' from any peak.

The approaches of both Schirmer et al. (2007) and Gavazzi & Soucail (2007) are based on complete photometric surveys of their fields. They evaluate the significance of cluster candidates internal to their surveys.

We follow a similar approach with a complete redshift survey. Because the redshift survey is complete we can sample the survey itself to evaluate the significance of galaxy condensations along the line of sight to lensing peaks. The redshift survey is shallower than the corresponding photometric surveys. We demonstrate below that redshifts for the more luminous galaxies in condensations at redshifts from 0.2 to 0.53 are adequate to demonstrate correspondence (or lack of it) between the weak-lensing peaks and clusters of galaxies in the redshift survey. We briefly explore the dilution of a "cluster" signal in photometric redshift data in Section 5. The sensitivity plot (Figure 1) suggests that we need not look deeper than z ∼ 0.55 to identify clusters corresponding to the weak-lensing peaks; more distant systems must be very rich and they should be obvious in the photometric data. Kubo et al. (2009) find no evidence for such higher redshift systems.

Probes through the redshift survey centered on each galaxy in the survey contain all of the selection effects which impact a probe toward a lensing peak. Thus, comparison of these probes with similar probes toward the peaks should provide a robust measure of the significance of cluster candidates in the redshift survey. We have demonstrated by experiment that centering of probes on galaxies or on random points in the region has no effect on the results of our analysis.

The steps in our candidate cluster identification procedure are as follows.

• 1.  
  We sample the SHELS survey complete to R = 20.3 in test cones with 3' radius centered on each galaxy in the complete redshift survey. The cones are large enough to detect a galaxy cluster across the redshift range we sample.
• 2.  
  In each test cone we count the number of additional galaxies, N_gal, within a bin of 1600(1 + z) km s⁻¹ centered on the survey galaxy. The bin size is comparable with the extent in redshift space of systems we wish to identify. The bin width is conservatively large.
• 3.  
  In each of the redshift bins of item 2, we evaluate the mean occupation and the variance (σ_SH) across the entire survey. We then identify the set of test cones at each redshift which are 5σ_SH above the mean occupation and contain at least six galaxies (N_gal ⩾ 5). The minimum number of galaxies enables computation of a dispersion (albeit with large error). The 5σ_SH limit restricts the sample to high peaks which are reasonable candidate clusters especially at the peak sensitivity of the survey, z ∼ 0.3. Both the six galaxy and 5σ_SH limits are generous; they admit many peaks well below the expected detection threshold for the weak-lensing map (Figure 1). We call these probes 5σ_SH probes hereafter.

Figure 4 shows a map of the F2 region. Each gray point marks the position of a galaxy at the center of a 5σ_SH probe. As a basic test of the efficacy of these probes in identifying known clusters of galaxies, the open circles in Figure 4 show that the locations of four known X-ray clusters, XMMU J091935+303155, CXOU J092026302938, CXOU J092053+302900, and CXOU J092110+302751, coincide with highly ranked probes. The cluster XMMU J091935+303155 is undersampled in the redshift survey because a saturated bright star (R ≲ 18) is superposed near its center. We comment further on these clusters in Section 3.4.

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Kubo et al. (2009) find 12 peaks in the convergence map with signal to noise ν ⩾ 3.5. The numbers in Figure 4 are centered on the coordinates of these significant weak-lensing peaks. Table 1 of Kubo et al. (2009) lists the coordinates and value of ν for these peaks. The significance of these peaks is similar to the peaks considered in other convergence maps and in the simulations by Hamana et al. (2004).

In Figure 4, the two most significant weak-lensing peaks are coincident with the two X-ray clusters, CXOU J092026+302938 and CXOU J092053+302900. These two clusters coincide with the cluster A781. All of the weak-lensing peaks except 4, 6, 8, and 12 lie within 3' of the center of at least one well-populated probe.

There are obviously many well-populated probes without any associated significant weak-lensing peak. Many of the well-populated SHELS probes correspond to groups at a redshift ≲0.3. From Figure 1, we would not expect these systems with σ_los ≲ 500 km s⁻¹ to produce a significant signal in the κ map.

To further assess the meaning of the weak-lensing peaks, we examine the redshift distributions along the line of sight toward each of the significant peaks. We ask which lines of sight intersect a cluster with a velocity dispersion large enough to plausibly account for the weak-lensing peak (Section 3.3). We also investigate the coincidence of SHELS systems with lower significance convergence peaks with 1<ν < 3.5 (Section 3.3). Finally, we check all of the 5σ_SH peaks in Figure 4 to see whether any others contain systems with large enough velocity dispersion that they should be detected according to the sensitivity plot (Figure 1; Section 3.5). We can then evaluate the completeness of the DLS cluster detections (see Sections 4 and 6).

In this section, we evaluate candidate systems along the line of sight toward weak-lensing peaks. We begin with the 12 most significant DLS peaks. We extend the discussion to all DLS peaks with ν ≳ 1 in Section 3.5.

Figures 5–8 show the redshift, z, distributions of galaxies within 3' probes centered on each of the 12 highest significance DLS peaks (see Figure 4). The bins are 800(1 + z) km s⁻¹. The dark histogram shows galaxies with R ⩽ 20.3; the gray histogram shows galaxies with R ⩽ 20.6. The histograms fall in three categories: (1) there is an obvious single peak which probably corresponds to a system of galaxies; (2) there is more than one apparently significant peak; and (3) there are no significant peaks.

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The probes toward peaks 1, 2, and 5 each contain an impressive peak. These peaks correspond to clusters of galaxies; we evaluate the dispersions in Section 3.3.

Peak 3 contains several well-populated peaks at redshifts 0.28–0.34. Peaks 4, 6, 8, and 12 contain no well-populated peaks. If there are condensations in redshift space near these peaks, most of the galaxies are more than 3' from the position of the weak-lensing peak. With a larger search radius of 4', Kubo et al. (2009) identify candidate systems associated with convergence map peaks 8 and 12.

Figure 9 shows higher resolution histograms for weak-lensing peaks 3 and 5 in both 3' and 6' probes. The upper histograms show the distributions for the magnitude limited samples with R ⩽ 20.3; the lower histograms show all of the galaxies with R ⩽ 20.6. The difference in the redshift distributions is obvious particularly in the better sampled histograms. Peak 5 is a cluster with a rest-frame velocity dispersion of 729 ± 41 km s⁻¹ (3' probe); the line of sight toward peak 3 contains a superposition of three peaks with rest-frame velocity dispersions of 150, 260, and 340 km s⁻¹.

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A detailed model is necessary to assess whether the superposed low mass systems along the line of sight toward peak 3 can account for the weak-lensing signal. This detailed discussion is beyond the scope of this paper; for the rest of the discussion we focus on lines of sight which include a plausibly rich cluster. We concentrate on evaluating the weak-lensing map as a method of identifying these rich systems.

The lines of sight toward peaks 7 and 9 each contain a peak populated by ∼10 galaxies. Peak 10 has a peak at 0.185 along with a peak at 0.53. The redshift survey is sparse at z ∼ 0.5; thus the number of galaxies in the peak at 0.53 is still indicative of the presence of a system (see Section 3.3). The fainter galaxies obviously enhance the peak. The situation is similar for the line of sight toward peak 11; there is a peak at z = 0.53 again enhanced by the fainter galaxies (see Section 3.4).

The apparent systems which appear as peaks in the redshift survey have a large range in velocity dispersion. We next determine which of these peaks correspond to systems with velocity dispersion (mass) sufficient to account for the weak-lensing signal.

In this section, we compute velocity dispersions for systems within our 3' probes. To construct a complete catalog of massive systems which should be detected by the DLS at ν ⩾ 3.5, we consider all apparent systems with rest-frame line-of-sight velocity dispersion σ_los,3 ≳ 500 km s⁻¹ measured from the R ⩽ 20.3 catalog. The velocity dispersion limit lies significantly below the v = 3.5 threshold in Figure 1. We also examine weak-lensing peaks with statistical significance ν ≳ 1 to examine the overall signal in the weak-lensing map and to evaluate the frequency of accidental superpositions between weak-lensing peaks and clusters in SHELS.

We use SHELS redshifts for galaxies with R ⩽ 20.6 to compute the rest-frame velocity dispersion, σ_los,3, for each of the candidate systems within a 3' probe centered on the position of the weak-lensing peak. We also compute the rest-frame velocity dispersion for each candidate system within a 6' probe concentric with the 3' probe. We make this comparison because the determination of the velocity dispersion of the clusters which comprise the A781 complex is problematic as a result of system overlap in redshift space. In general, the comparison of these two apertures also gives a measure of the systematic error in the determination of the velocity dispersion.

Table 1 lists the candidate systems and their properties for both a 3' and a 6' radius centered at the listed R.A.₂₀₀₀ (Column 4) and decl.₂₀₀₀ (Column 5). The table includes systems along the line of sight toward weak-lensing peaks with ν>1 as well as candidate SHELS systems where there is no corresponding peak in the convergence map. Table 1 lists the candidate cluster name (Column 1), the rank of the corresponding DLS peak if there is one (Column 2), the ν of the weak-lensing peak (Column 3), the mean redshift of the SHELS candidate system (Column 5), the rest-frame line-of-sight velocity dispersion within the 3' radius, σ_los,3 (Column 7), the number of survey galaxies in the 3' system, N₃ (Column 8), the bootstrap error in σ_los,3, err₃ (Column 9), the rest-frame line-of-sight velocity dispersion within the 6' radius, σ_los,6 (Column 10), the number of survey galaxies in the 6' system, N₆ (Column 11), and the bootstrap error in σ_los,6, err₆ (Column 12). For weak-lensing peaks 1 and 2, the table lists only 6' quantities. We discuss this issue in detail below. In all other cases, we limit the range where we compute the velocity dispersion with standard 3σ clipping. For nearly all of the cluster candidates (regardless of a counterpart in the weak-lensing map), the rest-frame velocity dispersions in the two apertures differ by ≲2σ. In fact, the only exception is DLS peak 15 where the difference is 2.2σ.

The abundance of systems of galaxies increases steeply with decreasing line-of-sight velocity dispersion. We limit our discussion of candidate lensing systems to massive halos with σ_los,3 ≳ 500 km s⁻¹. Coincidences of weak-lensing peaks with individual much less massive systems are common, as are superpositions along the line of sight. Of course, the greater abundance of these systems implies a larger chance of mismatching halos with weak-lensing peaks (see Hamana et al. 2004). Analysis of this issue is beyond the scope of this paper.

For the range of candidate cluster redshifts in Table 1, the 3' radius corresponds to 0.38 Mpc (z = 0.12) and to 1.1 Mpc (z = 0.53). These radii are within r₂₀₀, the radius where the enclosed average mass density, ρ(<r)₂₀₀ = 200ρ_c. Here, ρ_c is the critical density. The radius r₂₀₀, a proxy for the virial radius, ranges from 1.0 to 2.0 Mpc for well-sampled clusters with rest-frame line-of-sight velocity dispersions in the range 500–1000 km s⁻¹ (Rines et al. 2003; Rines & Diaferio (2006). According to the scaling relations of Rines & Diaferio (2006), the corresponding range of masses, M₂₀₀, within r₂₀₀ is 1.2×10¹⁴–1.2×10¹⁵ M_☉.

Rines & Diaferio (2006) show that the velocity dispersion in a cluster decreases with radius. For clusters at larger redshift, the tendency to underestimate the velocity dispersion in our physically larger aperture is compensated by the increasing difficulty of eliminating velocity outliers in the sparser samples.

Determination of the central velocity dispersion for weak-lensing peaks 1 and 2 is problematic because they overlap substantially in redshift space. Geller et al. (2005) published rest-frame velocity dispersions of 741⁺³⁵₋₄₀ km s ⁻¹ and 674⁴³₋₅₂ km s⁻¹ for peaks 1 and 2, respectively, based on identification of cluster members in a friends-of-friends group-finding algorithm applied to less complete SHELS data. Here, we explored the region of the A781 complex by first assessing the global redshift distribution within a 30' region centered midway between the coordinates listed in Table 1 for these systems. There are two well-defined peaks centered at redshifts 0.2915 (peak 1) and 0.3004 (peak 2) in essential agreement with the mean redshifts for these systems in Geller et al. (2005). Application of the routine mrqmin (Press et al. 1992) returns velocity dispersions for peaks 1 and 2 that substantially underestimate the central velocity dispersions because they sample well into the cluster infall region (Rines et al. 2003; Rines & Diaferio 2006). The mean redshifts are, however, robust and we use them to compute the velocity dispersion in the central regions of the clusters.

To estimate the central rest-frame velocity dispersion of peaks 1 and 2, we fix the mean redshift at the values obtained for the global sample (Table 1). For samples within a 3' radius of each center, a fit of two Gaussians in the Press et al. (1992) routines does not converge. Within a 6' radius (a physical radius of ∼1.5 Mpc, roughly the expected r₂₀₀ for these clusters) around each of the two centers, the fits converge to the rest-frame dispersions in Table 1. The error in Table 1 is an approximate and conservative estimate; the errors are dominated by systematic issues in separating the two clusters in redshift space. This approach yields systematically larger velocity dispersions than the group-finding algorithm used in Geller et al. (2005), but the differences are within the large error. Regardless of the method of assessing the velocity dispersion of these clusters, they both exceed the ν = 3.5 threshold (Figure 1).

There are 10 SHELS candidate systems along the line of sight toward weak-lensing peaks with ν>1. In one case, weak-lensing peak 10, there are two systems along the line of sight, one at z = 0.18 along with a more massive system at z = 0.53. The median separation between the lensing peak positions and the centers of the candidate galaxy cluster is 05 in agreement with the similarly small separations found in previous surveys (Gavazzi & Soucail 2007; Schirmer et al. 2007). Figure 10 shows the DLS weak-lensing peak rank as a function of statistical significance.

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Solid dots in Figure 10 represent cases where SHELS reveals a cluster along the line of sight to the peak with a rest-frame velocity dispersion ≳500 km s⁻¹ and z < 0.55. It is interesting to note that, as expected, the fraction of "cluster detections" (solid dots) is larger for ν_DLS ⩾ 3.5 than at lower significance.. In Section 4, we estimate the accidental coincidence rate and show that there is some signal in the weak-lensing map at ν_DLS < 3.5 even though, as expected, it is diluted by an increasing abundance of noise peaks as ν_DLS decreases. In contrast with Kubo et al. (2009), we find only four (rather than 10) convincing coincidences between rich clusters and weak-lensing peaks with ν ⩾ 3.5. The main reasons for the difference between our results and Kubo et al. (2009) are (1) our use of the redshift survey to compute line-of-sight velocity dispersions and (2) our use of a smaller search radius (consistent with the small offsets between lensing peak positions and galaxy system centers observed by other investigators from both data and simulations). Kubo et al. (2009) count superpositions like peak 3 as weak-lensing detections; we count only candidate systems which have a large enough SHELS line-of-sight velocity dispersion to exceed the ν = 3.5 threshold. We conclude that the weak-lensing detection efficiency for ν ⩾ 3.5 is 33%.

Chandra and XMM observations cover 14% of the DLS field with widely variable sensitivity. As a result of these observations, there are seven known extended cluster X-ray sources in the field (Figure 11). Both Chandra and XMM observations target the region near A781 which, remarkably, contains four cluster X-ray sources (Table 2 and Figure 11). In order of right ascension, Table 2 lists the X-ray cluster name (Column1), the rank of any corresponding DLS peak (Column 2), the R.A.₂₀₀₀ and decl.₂₀₀₀ (Columns 3 and 4), the rest-frame line-of-sight velocity dispersion determined from the SHELS galaxies in the system with R < 20.6 in a 3' cone (except for CXOU J092026+302938 and CXOU J092053+302900 where we quote the 6' dispersions from Table 1) centered on the X-ray position (Column 5), the number of galaxies in the system (Column 6) the bootstrap error in the velocity dispersion (Column 7), and the Chandra X-ray flux from the literature (Column 8). Double line entries for a single X-ray system indicate superposed structure in SHELS along the line of sight.

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The two clusters CXOU J092026+302938 and CXOU J092053+302900 are at redshifts of 0.3004 and 0.2915, respectively; these two clusters correspond to the original A781. They are also responsible for the two most significant weak-lensing peaks in the Kubo et al. (2009) map. The cluster CXOU J092026+302938 is complex; CXOU J092053+302900 is a simpler system (Wittman et al. 2006; Sehgal et al. 2008).

Khiabanian & Dell'Antonio (2008) developed the lensing map reconstruction technique applied by Kubo et al. (2009). In addition to the map with a uniform 15 resolution, Khiabanian & Dell' Antonio (2008) show a map with a resolution of 09 in the regions around A781. This map reveals a detection of a third X-ray cluster CXOU J092026+302938 and CXOU J092053+302900 with a S/N∼5.7. SHELS shows that this cluster has a mean redshift of 0.43 and a velocity dispersion of 754 ± 92 km s⁻¹ (Table 2).

The fourth extended cluster X-ray source in the region of A781, XMMU J091935+303155, is a puzzling and complex case. There is a well-defined, blue arc associated with the first-ranked galaxy located at R.A.₂₀₀₀ = 9:19:35.063, decl.₂₀₀₀ = 30:31:56.627, very close to the X-ray center. Our attempt to measure a redshift for the arc with the Blue Channel spectrograph on the MMT resulted in detection of a blue continuum but no emission lines. Thus, the redshift of the arc is probably ≳1.2. The mean cluster redshift, 0.43, is essentially the same as the redshift of the first-ranked galaxy (0.4276). This cluster is undersampled in SHELS because a region very close to the cluster center is excised around saturated bright star (R ≲ 18).

Although the velocity dispersion and redshift of XMMU J091935+303155 should put it above the detection threshold for the weak-lensing map, there is no significant peak associated with this system. The XMM observations (Table 6 of Sehgal et al. 2008) show that XMMU J091935+303155 has about twice the X-ray flux and nearly the same temperature as CXOU J092110+302751. CXOU J092110+302751 is at the same redshift and is cleanly detected in the higher resolution DLS weak-lensing map (Khiabanian & Dell'Antonio 2008). In contrast, we find only a ν = 0.5 enhancement in the convergence map at the position of XMMU J091935+303155. Sehgal et al. (2008) make an a posteriori model fit to the convergence map based on the X-ray data and argue for a weak detection. Both the SHELS data and the XMM observations suggest that the cluster should appear at a ν>3.5 in the convergence map.

In separate deep imaging with Chandra, Wittman et al. (2006) discovered three more extended X-ray sources: CXOU J091551+293637, CXOU J091554+293316, and CXOU J091601 +292750 (see Table 2 and Figure 11). In Figure 4, possible optical counterparts of CXOU J091551+293637 and CXOU J091554+293316 appear as populated 5σ_SH probes. CXOU J091551+293637 and CXOU J091554+293316 are adjacent to weak-lensing peak 11 and 8, respectively. These systems do not appear in Table 1 because the velocity dispersion for samples centered on the weak-lensing peaks and/or for samples limited to R < 20.3 fail our minimum cut of 500 km s⁻¹.

Table 2 lists line-of-sight velocity dispersions for condensations of galaxies along the line of sight toward these three extended X-ray sources. For CXOU J091551+293637, the probable first-ranked galaxy in the X-ray emitting system is within 30'' of the X-ray center and has a redshift of 0.53. We thus identify the extended X-ray source as a rich group at z = 0.53 with a rest-frame line-of-sight velocity dispersion of 360 ± 70 km s ⁻¹. The first-ranked galaxy and the 10 other probable members of the system are all fainter than R = 20.3. Thus, foreground structure at z ∼ 0.18 is responsible for the 5σ_SH probes in Figure 4. In Table 2, we quote a velocity dispersion for this foreground structure that is merely the spread of redshifts within the large-scale structure intercepted by our probe.

The X-ray source CXOU J091554+293316 has the lowest flux of the three (Wittman et al. 2006). There is no convincing optical counterpart; the nearest galaxy is an arcminute away from the X-ray center and only seven other galaxies within our 3' probe appear clustered at the z = 0.18; a normal X-ray emitting group at this redshift should be easily detectable with many members in a deep survey like SHELS. We list an apparent velocity dispersion in Table 2, but this dispersion is again a measure of the spread of redshifts within the large-scale structure intercepted by our probe.

The center of the southernmost extended X-ray source, CXOU J091601+292750, is within 10'' of a galaxy at z = 0.53. An additional six galaxies at similar redshift yield an estimate of the rest-frame line-of-sight velocity dispersion for this system of 523 ± 71 km s⁻¹. The foreground structure seen in the lines of sight toward CXOU J091551+293637 and CXOU J091554+293316 is also present here. Again the velocity dispersion for this structure at z = 0.18 is indicative of the spread of redshifts within the foreground large-scale structure. There is no weak-lensing peak near this X-ray system.

The rest-frame velocity dispersions for all three systems (CXOU J091551+293637, CXOU J091554+293316, and CXOU J091601+292750) should place them below the ν = 3.5 detection threshold for the weak-lensing map (Figure 1). Kubo et al. (2009), however, associate these clusters with weak-lensing peaks 8 and 11. It is conceivable that foreground structure at z = 0.18 and overlap among the clusters boosts the weak-lensing signal; detailed modeling beyond the scope of this paper is necessary to understand whether these superpositions are adequate to explain the apparent weak-lensing detection.

Among the seven known X-ray clusters in the DLS fields, only three (CXOU J092026+302938, CXOU J092053+302900, and CXOU J092110+302751) are cleanly detected as peaks with ν>3.5 in the convergence map provided the resolution is adequate. The cluster XMMU J091935+303155, detected by XMM and SHELS but not by the DLS, is puzzling. Another three clusters at greater redshift (and lower rest-frame velocity dispersion) may be detected in the DLS map as a complex superposition; the SHELS velocity dispersions and the sensitivity curve for the convergence map suggest that none of these systems should be detected on their own.

This analysis of X-ray observations of a small portion of the DLS field demonstrates some problematic issues for construction of catalogs of clusters of galaxies from weak-lensing maps: (1) coarse resolution of the weak-lensing map dictated by the density of sources can reduce the number of clusters in an uncontrolled way; (2) superpositions may increase the number of apparent detections in a complex way; and (3) inconsistencies among different cluster detection methods need to be understood both from simulations and from larger observational samples where direct comparison is possible. Bergé et al. (2008) are, for example, undertaking such an observational project in the X-ray.

SHELS obviously enables identification of cluster candidates independent of the weak-lensing map. There are many reasonable approaches to the identification of candidates. For consistency, we use the 3' probes of Figure 4. Other algorithms including the identification of systems with a friends-of-friends algorithm yields very similar results for these dense, well-populated systems (M. Ramella et al. 2010, in preparation).

To construct a catalog of cluster candidates for z < 0.55, we start with the probes shown in the map of Figure 4. We first identify all probes with at least six galaxies with R ⩽ 20.3 (significance ≳5σ_SH) in any redshift interval, 1600(1 + z) km s⁻¹, and with apparent velocity dispersion ≳500 km s⁻¹. We then examine all overlapping probes and choose the most populated to represent the system. We choose the center of this probe as the indicative center of the system.

The probes in Figure 4 are based solely on galaxies with R < 20.3. We next include the additional galaxies with 20.3 < R ⩽ 20.6. The broad window we use to construct the map may admit several groups with small velocity dispersion as in peak 3. Increased sampling enables elimination of several systems where we can identify better defined multiple peaks as convincing superpositions. We list all of the SHELS cluster candidates in Table 1. For completeness, we include all of the candidate clusters we find with line-of-sight velocity dispersion ≳500 km s⁻¹ even though many of these should not be detected at high significance in the DLS weak-lensing map. For redshifts ≲0.4, we easily identify clusters with line-of-sight velocity dispersion ∼500 km s⁻¹; for larger redshift the lowest velocity dispersion system has σ_los,6 = 642 ± 70 km s⁻¹.

The sample of SHELS candidate clusters with velocity dispersions large enough to be detected in the DLS weak-lensing map should be complete (see Figure 1). The set of 20 SHELS clusters (Table 1) includes 10 systems along the line of sight toward weak-lensing peaks with ν ≳ 1. We also find the two additional X-ray clusters SHELS J0921.2+3028 (CXOU J092110+302751) and SHELS J0919.6+3032 (XMMU J091935+303155). A higher resolution weak-lensing map detects the first of these; there is no weak-lensing detection of the second (see Section 3.4). Considering the errors in the line-of-sight velocity dispersion, σ_los,6, the SHELS candidate cluster list includes eight clusters with line-of-sight velocity dispersion large enough that they should be detected with ν ≳ 3.5 in the DLS map at the resolution we explore here; only four of them are detected (we exclude J0921.2+3028 which requires a higher resolution map for DLS detection). All four of the undetected clusters are at a redshift z ∼ 0.4. It is interesting that three of these clusters (two at z ∼ 0.4) are along the line of sight toward DLS peaks with 1.6 ⩽ ν < 2.9. Details of the cluster structure, voids along the line of sight, and noise in the κ map may all contribute to reduction of the weak-lensing signal (Hamana et al. 2004). Our data do not allow discrimination among these possibilities.