PROPERTIES OF WEAK LENSING CLUSTERS DETECTED ON HYPER SUPRIME-CAM's 2.3 deg² FIELD

DLS Field 2 was observed on 2013 February 4, Hawaii time. As is shown in Table 1, two pointing centers are chosen whose angular distance is 075 apart. Because the field of view of HSC is 15, we have substantial overlap between two pointings. The exposure time is either 300 s (nine exposures on south, eight exposures on north) or 150 s (four exposures on north), and the total exposure time of south and north pointing is 2700 and 3000 s, respectively. A circular dithering pattern of radius 2 arcmin is adopted around each pointing center to fill the gap of CCDs. The position angle is increased by 72°–90° between exposures.

Table 1.  Exposure Information

Field	Filter	J2000	${{T}}_{\mathrm{exp}}$ (s)	Med. Seeing ('')
South	HSC-i	(139.50, 30.00)	2700	0.58
North	HSC-i	(139.50, 30.75)	3000	0.57

Download table as:  ASCIITypeset image

The median stellar image (PSF) size of each exposure varied between 0 52 and 0 63 and the overall average was 0 58. Figure 1(a) shows the map of the PSF size over the FOV. The non-uniformity is visible and we observe a hump of 0 65 in lower left bottom and a minimum of 0 51 at slightly upper-right from the center. This is not expected from the design and the tolerance analysis and suggests that at this early stage the collimation of the optical system was insufficiently accurate.

To investigate this, we evaluate the spatial variation of the ellipticities of stars, $\{{e}_{1},{e}_{2}\}$, which is defined as

where I_ij is the Gaussian-weighted quadrupole moment of the surface brightness distribution.

Figure 1(b) shows a whisker plot where we present the position angle of the ellipse with a bar whose length is proportional to the ellipticity. We find a typical pattern of astigmatism generated by the tilt of the camera optics with respect to the optical axis of the primary mirror. In this case, the camera is tilted around the axis shown in the figure as a dashed line. From the start to the end of this observation, the instrument rotator (InR) rotated by 68°. The fix pattern seen on the FWHM and whisker plot co-rotated as the InR rotated, which supports of our interpretation of the non-uniformity of the PSF.

Recall that these test images were taken in the very first engineering run of the full FOV; we were trying to establish the way to measure the collimation error. Thus these particular images were taken under rather poor alignment conditions. The median ellipticity over the FOV is roughly 5% (Figure 1(b)), which is slightly larger than the typical 3%–4% raw ellipticity seen on Suprime-Cam images (Miyazaki et al. 2007; Figure 1). It is hard to determine the inclination angle from this data set alone but it is roughly a few arcmin. At the time of writing, we have established a way to measure the tilt and have implemented an auto-focus system, and we no longer see non-uniform patterns like this (S. Miyazaki et al. 2015, in preparation).

Zoom In Zoom Out Reset image size

HSC has 116 2k × 4k CCDs in total: 104 for imaging, in the central part of the field, and at the edges 4 CCDs for auto-guiding and 8 CCDs for focus monitoring. We use the auto-guiding for the observations reported here. Each CCD has 2048 × 4176 physical pixels (15 μm: 0 169 at the field center). The imaging area of each device consists of 4512 × 4176 stripes; each stripe is read through an independent output amplifier located on one side of one of the shorter edges. Each segment has an 8 pixel wide pre-scan and we add 16 pixels of over-scan along the serial register and 16 lines of over-scan along the parallel register.

We use the serial over-scan region for the zero reference of each amplifier (bias level). The median value of the pixel data on one row is used for the bias level for pixels on that row and subtracted. Non-physical reference pixels are trimmed from each segment and the segments are tiled together to reconstruct the original 2048 × 4176 image. We then divide the image by a flat field image that is generated from an average of dome flatfield images (typically ten images). Sky level is evaluated by medianing on a mesh whose element size is 256 × 256 pixels and fitted with a 2D polynomial. This is subtracted from the image. This reduced CCD image file is one unit of the data source for the shape measurement (Section 2.3).

The basic image analysis pipeline is developed through a collaboration of multiple institutions including Princeton University, Kavli IPMU, and NAOJ. It is based on a customized version of the "LSST-stack" (Ivezić et al. 2008; Axelrod et al. 2010), a software suite being developed for the LSST project. Object detection is made on each CCD image. The PSF on a CCD is modeled as a function of the CCD pixel coordinates, and can be reconstructed accurately at any pixel position. This is used in the subsequent morphological classification process which tags star/galaxy flags as well as flagging cosmic rays. The flags are stored on the mask layer in an output FITS file.

The star catalog is correlated with an external catalog (SDSS-DR8) for each CCD to perform photometric and astrometric calibration, by using a cross-matching algorithm formulated by Tabur (2007). The external catalog is prepared in the form of indexed files of astronomy.net for fast access to the source information. The SDSS magnitudes of the cross-matched sources are transformed into the very similar HSC bands to derive photometric zeropoints, and the SDSS coordinates are fit to determine the world coordinated system of each CCD.

Because the number of stars on each CCD matched with the astrometric catalog is limited (≤50), the accuracy of the astrometric solution is not sufficient for the later image stacking process. In order to minimize the mosaic-stacking error, the residual of the ith control star on the eth exposure from a reference frame, $({\rm{\Delta }}{x}_{{ie}},{\rm{\Delta }}{y}_{{ie}})$, is parameterized as a polynomial function of the field position, $(x,y)$, as

$$\begin{eqnarray}&&{\rm{\Delta }}{x}_{{ie}}=\underset{l=0}{\overset{N}{\displaystyle \sum }}\underset{m=0}{\overset{l}{\displaystyle \sum }}{a}_{{lme}}{x}^{l-m}{y}^{m}\ \ \ \ {\rm{\Delta }}{y}_{{ie}}=\underset{l=0}{\overset{N}{\displaystyle \sum }}\underset{m=0}{\overset{l}{\displaystyle \sum }}{b}_{{lme}}{x}^{l-m}{y}^{m}.\end{eqnarray} \tag{ 2 }$$

This is a "Jelly CCD" approach adopted in Miyazaki et al. (2007), which is particularly important for HSC where we have a significant non-axisymmetric distortion pattern caused by the larger index mismatch of the glass used for the atmospheric dispersion corrector (S. Miyazaki et al. 2015, in preparation). The implementation on the HSC pipeline is more sophisticated. Rather than dealing with the polynomial as a function of the pixel coordinate as above, the SIP convention (TAN-SIP) of the World Coordinate System (WCS) is employed to allow the Jelly warp of the CCDs. A point on Figure 2(a) shows the displacement of the astrometric position of the control star with respect to the position on the first exposure; thus the scatter presents a measure of the mosaic-stacking error. This internal error is as low as 6–7 mas. A point on Figure 2 (b) shows the displacement from the external catalog and the scatter is larger (abut 60 mas), which we suppose is partly due to the astrometric error of the external catalog itself. The peak position of the distribution in (b) has no offset from the origin, which indicates that there is no systematic error on the astrometric solution. The combination of the internal (a) and external (b) error gives an estimate of the total astrometric error of ∼20 mas in this analysis. Once the global solution over multiple exposures is obtained, the WCS of each CCD image is refreshed for further usage in the lensing analysis.

Zoom In Zoom Out Reset image size

Handling an entire mosaic-stacked image all at once is usually not practical. We divide the image into a set of patches whose size is $4200\times 4200$ pixel. In order to take care of objects that fall on the boundary of the patches, we implement 100 pixel overlap between neighboring patches. The patch is a unit comprising the following mosaic-stack operation and the object detection on the co-added image. We then combine the object catalogs of all patches into a single catalog by removing the objects detected multiple times within the overlapped region. From the combined catalog, we extract the coordinates of moderately bright stars (22.0 < HSC-i < 24.5) for the star catalog and the coordinates and the magnitude of galaxies. The star and the galaxy catalogs as well as the image files with refined WCS of each CCD (before the stack) are handed over to the next stage where we carry out the weak lensing shear estimate.

In the Suprime-Cam weak lensing survey (Miyazaki et al. 2002a, 2007), we employed a rather traditional method where we measured the galaxy shapes on the fully reduced mosaic-stacked CCD image. The PSF is not precisely round, nor is it of uniform size over the whole field. It is evaluated from star images over the whole field. (Equation (1)). Because the PSF varies over the field of view (Figure 1(b)), the PSF ellipticities are represented as a polynomial function of the field position. Analysis on the mosaic-stacked image requires that the PSF pattern does not change much during the series of exposures and that the CCD boundary does not cause a discontinuity of the PSF variation. Otherwise, the PSF becomes too complicated to represent as a simple polynomial function of the field position, and could cause systematic error in the galaxy shape measurement. In the case of the Suprime-Cam survey, we determined that the conditions were mostly met because one field was observed in a relatively short time span (40 minutes) with a small (∼2 arcmin) dithering offset and we had a small CCD mosaic. However, in the planned HSC survey one field observation is split into several semesters to search for variables and the circular FOV requires a relatively large (nearly a half of the FOV) dithering step. It is therefore not expected that this simple requirement will be met. Instead, the galaxy shapes will be evaluated on the pre-stacked image and then combined after that to reduce the statistical error. When the survey is underway, we will develop code to determine shapes and fluxes simultaneously from the constituent images, which is arguably the most statistically efficient use of the data.

For these early data, we have adopted Bayesian galaxy shape measurements implemented in the "lensfit" algorithm (Miller et al. 2007; Kitching et al. 2008) in this work. The PSF employed in lensfit is not a simple elliptical but a more empirical 32 × 32 pixel postage stamp image. Each pixel value of the postage stamp is fitted independently on each exposure to a 2D-polynomial function of the sky coordinate. lensfit also allows varying low-order coefficients between CCDs to further minimize the residual on each CCD.

Galaxy shape is measured on the 40 × 40 pixel postage stamp image of each galaxy. From a galaxy coordinate on the given catalog, the postage stamp image is trimmed from the unwarped original CCD image by using the WCS information. Usually, the galaxy is imaged on multiple exposures and multiple postage stamps are created for one galaxy. lensfit has two galaxy model components: de Vaucouleurs profile bulge and exponential disk. The centers of the two models are assumed to be aligned. The ellipticity, the 1D size, the normalization, and the bulge-to-disk ratio are the parameters that model a galaxy. The postage stamp image of the model galaxy is convolved with the estimated PSF image at the galaxy position and the convolved image is compared with the observed galaxy image to calculate the residual. By minimizing the residual, the best-fit galaxy model and the ellipticity are estimated. In the fitting process, data from multiple exposures are simultaneously fitted. Note that the galaxy position is a free parameter as well, and is marginalized by integrating under the cross-correlation function of the galaxy model and the data. Therefore, the astrometric error in the mosaic-stack process has limited impact on the error in the shape measurement. Note also that the observed image is not resampled nor warped onto the new coordinate in this process, which avoids the introduction of correlated noise mentioned in Hamana & Miyazaki (2008).

For PSF modeling, we employ second order polynomials for the exposure-wide fit and a first order polynomial for the CCD term. Figure 3(a) shows the ellipticity whisker plot of one exposure that is calculated from the PSF model postage stamp and averaged in the grid to visually compare with the observed PSF shown in Figure 1. Figure 3(b) shows the difference in the observed and modeled ellipticity. The residual is sufficiently small as ${\sigma }_{e}\;\lt \;1$%.

Zoom In Zoom Out Reset image size

In the following sections, we adopt galaxies brigher than $i=24.5$ for the mass map reconstruction. The rms sigma of the ellipticity of adopted galaxies is 0.41 and the galaxy number density is 20.9 arcmin⁻². In order to estimate the shear bias, we calculate mean ellipticities of galaxies used for the lensing analysis. The result is $(\langle {e}_{1}\rangle ,\langle {e}_{2}\rangle )=(0.005,-0.004)$, which is sufficiently small for this study. However, the bias is not completely negligible for cosmic shear studies and further investigations about the origin will be necessary in the future.

The dimensionless surface mass density, the convergence $\kappa ({\boldsymbol{\theta }})$, is evaluated from the tangential shear as

$$\begin{eqnarray}&&\kappa ({\boldsymbol{\theta }})=\int {d}^{2}\phi {\gamma }_{t}({\boldsymbol{\phi }}:{\boldsymbol{\theta }})Q(| {\boldsymbol{\phi }}| )\end{eqnarray} \tag{ 3 }$$

where ${\gamma }_{t}({\boldsymbol{\phi }}:{\boldsymbol{\theta }})$ is the tangential component of the shear at position ${\boldsymbol{\phi }}$ relative to the point ${\boldsymbol{\theta }}$ and Q is the filter function. We adopt as a filter the truncated Gaussian

$$\begin{eqnarray}&&{Q}_{G}(\theta )=\displaystyle \frac{1}{\pi {\theta }^{2}}\left[1-\left(1+\displaystyle \frac{{\theta }^{2}}{{\theta }_{G}^{2}}\right)\mathrm{exp}\left(-\displaystyle \frac{{\theta }^{2}}{{\theta }_{G}^{2}}\right)\right],\end{eqnarray} \tag{ 4 }$$

for $\theta \;\lt \;{\theta }_{o}$ and ${Q}_{G}=0$ elsewhere (Hamana et al. 2012). We employ ${\theta }_{G}=1$ arcmin and ${\theta }_{o}=15$ arcmin, respectively.

We adopt 9 × 9 arcsec grid cells and then calculate the convergence, $\kappa ({\boldsymbol{\theta }})$, on each grid using Equation (3) to obtain the κ map. In order to estimate the noise of the κ map, we randomized the orientations of the galaxies in the catalog and created a ${\kappa }_{\mathrm{noise}}$ map. We repeated this randomization 100 times and computed the rms value at each grid point. Assuming the κ error distribution is Gaussian, this rms represents the 1σ noise level, and thus the measured signal divided by the rms gives the S/N of the convergence map at that point.

Figure 4 is the convergence S/N map so obtained. Wittman et al. (2006) showed a convergence map based on the DLS R-band imaging. We find general agreement with that map although the resolution of our map is higher. Note, however, that the convergence map of Wittman et al. (2006) was generated in the middle of the survey and it did not employ the full depth image of DLS. They identifed two shear-selected clusters on the map: DLSCL J0920.1+3029 (140.033, 30.498) and DLSCL J0916.0+2931 (139.000, 29.526), and we see the corresponding peaks on our map.

Zoom In Zoom Out Reset image size

The former is a complex Abell 781 cluster region where at least four clusters have been identified so far from X-ray data and spectroscopic follow-up observations (Sehgal et al. 2008). In Figure 5, we show a close-up view of the region. Crosses indicate the center of X-ray emissions (XMM) of the four clusters, which are called "west" (z = 0.4273), "main" (z = 0.302), "middle" (z = 0.291), and "east" (z = 0.4265) respectively from west to east. We detect clear corresponding peaks in the latter three clusters. This demonstrates that the angular resolution of the weak lensing convergence map matches well with that of the XMM X-ray image.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We see two separate peaks on the convergence map in the "main" cluster region and the mid-points of the two peaks roughly coincide with the location of the X-ray emission center. The X-ray image is mostly round and no corresponding structure that is found on the lensing map is observed (Sehgal et al. 2008). Wittman et al. (2014) presented the convergence map on this A781 region based on the full depth DLS imaging but no structure is seen in the "main" cluster in that map. The structure seen in Figure 5 survived several reality checks in our analysis (change of the magnitude cut of galaxies and bootstrap resampling of galaxies) but obviously further observations are necessary to confirm the structure. If it is confirmed, it will provide another laboratory for testing the nature of dark matter following the famous "bullet cluster." In fact, Sehgal et al. (2008) reported a small extended substructure to the south-west of the X-ray peak. Venturi et al. (2011) discovered possible a radio relic in their deep radio map. These are indicative of a merger and have already suggested that the "main" is not simple system.

No significant peak, however, is found at the "west" position in Figure 5. In fact, the lensing signal of the X-ray emitting cluster Abell 781 "west" has been a matter of debate. Motivated by the less significant detection on the original DLS map (Wittman et al. 2006), Cook & Dell'Antonio (2012) examined the region based on images independently obtained by the Orthogonal Parallel Transfer Imaging Camera on the WIYN 3.5 m telescope and Suprime-Cam. They concluded that no significant signal was observed on either of the convergence maps and argued that this example could pose a challenge to the usefulness of weak lensing for the calibration of lensing mass against other observable properties of clusters in observational cosmology. Wittman et al. (2014) claimed, however, that the strong "joint constraint" of Cook & Dell'Antonio (2012) was an artifact of incorrectly multiplying p-values. They revisited the DLS data and tried to eliminate the contamination of foreground galaxies by adopting photometric redshift (Photo-z) probability density weighting in the reconstruction of the convergence map. They suggested that the peak on the revised convergence map is still less significant (2.7σ level) but this does not strongly exclude the X-ray mass. We estimate the WL mass at the X-ray position of the "west" cluster and the result is shown in Table 3. It is lower than what was obtained by Wittman et al. (2014) but the 1σ errors overlap each other. Therefore, our result supports the summary by Wittman et al. (2014) saying that the mass of "west" cluster (${M}_{200}$) is in the range 1–3 $\;\times \;{10}^{14}{M}_{\odot }$.

The red symbols in Figure 4 show the location of clusters of galaxies registered on the NASA/IPAC Extragalactic database (NED). It is clear that the symbols tend to exist in the colored area (stronger lensing signal) and we see a general correspondence of the weak-lensing peaks and the cluster positions. Among them, the star and diamond symbols are SHELS clusters which are identified by Geller et al. (2010) employing uniform spectroscopic observation of a magnitude-limited (R < 20.6) sample using HectoSpec. Geller et al. (2010) matched their clusters with the DLS lensing peaks; the star symbols show the matched clusters and the diamonds those unmatched.

In order to make an independent comparison between the light and mass in this region, we search for clusters using the DLS public photometric data with a method, the "CAMIRA," developed by Oguri (2014). CAMIRA makes use of the stellar population synthesis (SPS) models of Bruzual & Charlot (2003) to compute SEDs of red-sequence galaxies, estimates the likelihood of being cluster member galaxies for each redshift using ${\chi }^{2}$ of the SED fitting, constructs a three-dimensional richness map using a compensated spatial filter, and identifies cluster candidates from peaks of the richness map. For each cluster candidate, the brightest cluster galaxy candidate is identified based on stellar mass and location. Readers are referred to Oguri (2014) for more details of the algorithm and its performance studied in comparison with X-ray and gravitational lensing data.

We apply the CAMIRA algorithm to the DLS BVRz-band data. We apply a magnitude cut of $R\lt 24.5$ to exclude galaxies with large photometric errors. In Oguri (2014), a number of spectroscopic galaxies in SDSS have been used to calibrate the SPS model. We adopt this SDSS calibration result, but we also add a constant 0.02 mag error quadratically to the model scatter, in order to accommodate the systematic offset of magnitude zeropoints and the difference in magnitude measurements between DLS and SDSS. We confirm that this SDSS-calibrated SPS model provides reasonable ${\chi }^{2}$ values when fitted to spectroscopic SDSS red-sequence galaxies in the DLS field. The cluster catalog contains, for each cluster, the cluster center based on the brightest cluster galaxy identification, the photometric redshift, and the richness. The redshft and richness ranges are restricted to $0.1\lt z\lt 0.8$ and $N\gt 10$, respectively. In addition, in this paper we also compute the total stellar mass by summing up weighted stellar mass estimates of individual galaxies; the weights are the "weight factors" ${w}_{\mathrm{mem}}$, which resemble the membership probability of each galaxy (see Oguri 2014). Stellar mass estimates of individual galaxies are obtained from the SPS model fitting in which we assume a Salpeter initial mass function.

We estimate the stellar mass only using an optical data set which could cause a significant error in the estimation because of the limited band-pass. When we discuss the cluster members in this work, however, we mainly deal with passive galaxies whose stellar mass is expected to be easier to estimate than that of other general galaxies. In fact, Annunziatella et al. (2014) argued that the stellar mass of passive galaxies estimated from optical data agrees well with what is estimated from optical and NIR data within 25% if they adopt the template of passive galaxies in the calculation. Therefore we expect that our stellar mass estimates are not significantly biased by the absence of NIR data.

In Figure 6, we show the positions of resultant optically selected clusters as open circles where the center of the clusters is defined as the position of the bright cluster galaxy (BCG). The redshift is color-coded as is presented on the side bar. The diameter of the circles is 3 arcmin.

Zoom In Zoom Out Reset image size

We have searched for peaks in the convergence map; the results are indicated on Figure 6 as filled triangles (significant peaks of S/N > 4.5) and filled squares (moderate peaks of 3.7 < S/N < 4.5). We then match the peaks with the centers of the optically selected clusters and count the coincidences as we vary the match tolerance. As is shown in Figure 7, when we increase the tolerance from zero, the number of matches rises rapidly and reaches a plateau at about 1.5 arcmin. The displacement of the BCG from the cluster center (as defined by the X-ray emission center) can be as large as 0.5 Mpc/h (Oguri 2014), which corresponds to ∼1.5 arcmin at the redshift of 0.7, the highest redshift in our sample. We would therefore expect to have to use a match tolerance of this order to recover real matches, and this is what we see; at approximately this tolerance we have recovered all the real matches and the number should plateau, as it does. The number of matches increases again slowly beyond ∼4 arcmin, which we understand as accidental coincidences.

Table 2 shows the list of our peaks sorted by the S/N of the convergence map. Among nine significant peaks of S/N > 4.5, five peaks (peak IDs 1, 3, 5, 6, 7) do not have a corresponding optically selected cluster using the CAMIRA algorithm described in the previous section. We carefully examine each case here.

To complement our CAMIRA catalog, we looked for the clusters of galaxies in the NASA/NED database; the matched clusters are shown in the last column of the table where the tolerance is set at 1.5 arcmin. Peak IDs 3 and 7 do have counterparts on NED (SHELS J0920.4+3030:z = 0.3004 and WHL J092104.1+303424:z = 0.2758, respectively). These are proxy of the prominent A781 main cluster (ID 0) and have similar redshift to the main cluster. In the CAMIRA algorithm, we eliminate the member galaxies of detected clusters to avoid double-counting. CAMIRA also uses a compensated spatial filter, which suppresses detection of clusters near very massive clusters. These effects would explain why the CAMIRA catalog failed to detect clusters at the position of shear peak IDs 3 and 7. The situation might be improved by modifying the form of the spatial filter for optical cluster finding but we leave this fine-tuning for future work.

Peak ID 5 matched with the original DLS shear-selected cluster: DLSCL J0916.0+2931 (Wittman et al. 2006) which is another complex system in this region. They reported that there are three associated X-ray peaks along the north–south line and the north and the south peaks were confirmed spectroscopically as clusters at a redshift of 0.53. We, in fact, note that below the richness threshold of 10 adopted here, there is an optically selected cluster at Photo-z of 0.542 (richness = 8.05) 1.5 arcmin north of peak ID 5. The central X-ray peak is later confirmed as a cluster at z = 0.163 (Geller et al. 2010).

Peak ID 6 is supposed be a possible substructure of peak ID 0 because the X-ray emission peaks at just between peak 0 and peak 6 (Figure 5).

We therefore conclude that, apart from peak ID 1 where no deep multi-color data are available because it is outside the DLS field, all the other very significant peaks of S/N > 4.5 are generated by physical entities.

At a somewhat lower S/N level, Miyazaki et al. (2007) identified 17 peaks with S/N over 3.7 in a 2.8 deg² region in the XMM-LSS field, and found that nearly 80% of the peaks have physical counterparts. On the other hand we have 26 peaks on the DLS overlapped 2 deg² region and only 50% of the moderately significant (S/N > 3.7) peaks have identified physical counterparts. The discrepancy might be partially explained by the different noise level caused by the conservative magnitude cut adopted in this work, which resulted in a smaller number density of weak lensing galaxies, which in turn raises the noise level on the convergence map. When we raise the threshold from 3.7 to 4.5 we see a better match on the DLS field: nine peaks out of ten have counterparts. Note that the number density of the matched peaks is quite similar on XMM-LSS (4.3 peaks deg⁻² for S/N 3.7) and DLS (4.0 peaks deg⁻² for S/N > 4.5).

We now examine how the optically selected clusters match with the peaks on the convergence map in a different way. In Figure 8, circles show the redshift and the richness of the clusters detected by the CAMIRA algorithm. The clusters matched with the most significant peaks (S/N $\geqslant$ 4.5) are marked with triangle symbols, whereas those matched with moderate peaks (3.7 < S/N < 4.5) are marked with squares. It is encouraging to know that all luminous cluster samples (richness > 32) have corresponding convergence peaks.

Zoom In Zoom Out Reset image size

A cluster (Photo-z = 0.29) that is just below the richness threshold and has no associated peak is SHELS J0918.6+2953 whose spectroscopically confirmed redshift is 0.3178. We notice that this is one of the shear-selected samples in Kubo et al. (2009; Rank = 5, ν = 3.9). Although we have a positive convergence signal (∼2.5) here we see no strong peak (S/N > 3.7). Also, we have a peak (S/N = 3.9) 5 arcmin north of SHELS J0918.6+2953. When we adopt a larger smoothing kernel (${\theta }_{g}$ = 2 arcmin) on the convergence map, the positive signal is connected with the northern peak and results in a more significant peak (∼3.7) which can be a counterpart of SHELS J0918.6+2953. In this paper, however, we will keep a single smoothing scale of 1 arcmin for easy comparison with theoretical expectations.

There is a cluster at Photo-z of 0.5204 whose richness is low (12) but is matched with a significant peak (ID = 8). This peak is reported in Utsumi et al. (2014; Rank 3) as well using a totally independent data set and data analysis pipeline: Suprime-Cam data analyzed by imcat. We therefore suppose that this is not a spurious peak caused by systematic errors. Although they observe a spatial concentration of eight galaxies at redshift 0.537 (Δz = 0.025), they did not identify the peak as a cluster because it did not meet the more stringent SHELS cluster criteria (Geller et al. 2010).

The masses of the clusters are estimated from the tangential alignment of the shear. The shear is azimuthally averaged over the successive annuli placed with a logarithmic interval in radius of 0.2. We fit the radial profile with a singular isothermal sphere (SIS) model (surface density $\propto \;{\theta }^{-1}$). In order to minimize dilution by the member galaxies of clusters (and any intrinsic alignment signal), we avoid the center; the fitting was made from the radius of 1.8 to 12 arcmin.

The tangential shear, ${\gamma }_{T}$, induced by an SIS model is

$$\begin{eqnarray}&&{\gamma }_{T}=\displaystyle \frac{1}{{{\rm{\Sigma }}}_{\mathrm{cr}}}\displaystyle \frac{{\sigma }_{\mathrm{SIS}}^{2}}{2G}\displaystyle \frac{1}{r}\end{eqnarray} \tag{ 5 }$$

where ${{\rm{\Sigma }}}_{\mathrm{cr}}$, ${\sigma }_{\mathrm{SIS}}$, and r are the critical surface density, the velocity dispersion of the SIS, and the radial distance from the center;

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{cr}}=\displaystyle \frac{{c}^{2}}{4\pi G}\displaystyle \frac{{D}_{S}}{{D}_{L}{D}_{{LS}}},\ \ r={D}_{L}\theta .\end{eqnarray} \tag{ 6 }$$

Here, D is angular diameter distance. By putting Equation (6) into Equation (5), we have

$$\begin{eqnarray}&&{\sigma }_{\mathrm{SIS}}^{2}=\displaystyle \frac{{c}^{2}}{360}\displaystyle \frac{{D}_{S}}{{D}_{{LS}}}{\gamma }_{T}(\theta =1[\mathrm{deg}]).\end{eqnarray} \tag{ 7 }$$

The mass of SIS within the radius ${r}_{{{\rm{\Delta }}}_{c}}$ is given by

$$\begin{eqnarray}&&M(\lt {r}_{{{\rm{\Delta }}}_{c}})=\displaystyle \frac{2{\sigma }_{\mathrm{SIS}}^{2}}{G}{r}_{{{\rm{\Delta }}}_{c}}.\end{eqnarray} \tag{ 8 }$$

The mass is also described by adopting the critical over-density parameter, ${{\rm{\Delta }}}_{c}$, with respect to the mean density, $\bar{\rho }(z)$, as

$$\begin{eqnarray}&&M(\lt {r}_{{{\rm{\Delta }}}_{c}})=\displaystyle \frac{4\pi }{3}{r}_{{{\rm{\Delta }}}_{c}}^{3}\bar{\rho }(z){{\rm{\Delta }}}_{c}(z){/{\rm{\Omega }}}_{M}.\end{eqnarray} \tag{ 9 }$$

Eliminating ${r}_{{{\rm{\Delta }}}_{c}}$ from Equations (8) and (9) yields

$$\begin{eqnarray}M(\lt {r}_{{{\rm{\Delta }}}_{c}}) & = & 4.16\;\times \;{10}^{14}{M}_{\odot }{h}^{-1}{({\sigma }_{\mathrm{SIS}}/1000[\mathrm{km}\;{{\rm{s}}}^{-1}])}^{3}\\ & & \times {\left(\displaystyle \frac{1}{{(1+z)}^{3}}\displaystyle \frac{500}{{{\rm{\Delta }}}_{c}}\right)}^{\displaystyle \frac{1}{2}},\end{eqnarray} \tag{ 10 }$$

where $\bar{\rho }(z)={(1+z)}^{3}{{\rm{\Omega }}}_{M}{\rho }_{\mathrm{cr}}^{0}$ is adopted. We fit the data to obtain ${\gamma }_{T}(\theta =1[\mathrm{deg}])$ and the mass is estimated by using Equations (7) and (10) where we employ the redshift distribution of the source galaxies, $n({z}_{s})$, from Le Fèvre et al. (2013) rather than adopting the Photo-z estimated only from optical BVRz data to avoid possible systematic effects from Photo-z outliers. Note that the weak lensing mass inferred from the SIS model does not differ from what is obtained from the NFW model in our fitting range. The difference depends on the mass but it is less than 10% in most cases, which is rather smaller than the statistical error.

Based on the comparison of the Canada–France–Hawaii Telescope (CFHT) MegaCam data with the GREAT simulation, Miller et al. (2013) argued that lensfit slightly underestimates the multiplicative factor, m, of the shear especially when the signal-to-noise ratio of the objects, ν, is low, i.e., $\nu \;\lt \;20$, whereas the underestimate becomes less than 5% when $\nu \;\gt \;30$. Although we have not completed a comprehensive comparison of HSC data with the simulation the behavior of lensfit on HSC data should not be significantly different from that of CFHT MegaCam because the image quality and pixel scale are similar. Therefore, we should have a level of roughly 5% underestimate of shear at maximum which results in a roughly 7% underestimate in the mass evaluation. Since this is small compared with the statistical error, we will not deal with the systematic error explicitly in this paper.

Table 3 shows the result of the comparison of our mass estimate of A781 components with the values in the literature (Wittman et al. 2014). Despite the totally independent observation and the analysis, the agreement is encouraging and implies some progress in the convergence of weak lensing data analysis techniques.

Table 3.  Comparison of the Mass Estimate (${M}_{200}/{10}^{14}{M}_{\odot }$)

A781	This Work	Wittman et al. (2014)
Main	${7.0}_{-1.6}^{+1.8}$	${6.7}_{-1.3}^{+1.4}$
Middle	${4.6}_{-1.2}^{+1.4}$	${4.3}_{-1.2}^{+1.6}$
East	${4.8}_{-1.3}^{+1.6}$	${2.8}_{-1.2}^{+1.9}$
West	${1.2}_{-0.5}^{+0.7}$	${2.7}_{-1.0}^{+1.5}$

Note. The error is 1σ. Note that the west cluster mass is estimated at the X-ray peak whereas all the other masses are estimated at the weak lensing peak position.

Download table as:  ASCIITypeset image

Figure 9(a) shows the relation between the richness and cluster virial mass, ${M}_{500}$, of the optically selected cluster samples that have counterparts with the convergence peaks. The correlation is clearly seen and the slope is roughly 1–1.5, which agrees nicely with the slope found in Oguri (2014) where the richness–mass relation was derived from stacked weak lensing analysis with the CFHTLenS shear catalog. This further supports the reality of the cross-match of our lensing peaks and the optically selected clusters.

Zoom In Zoom Out Reset image size

In Figure 9(b), we show the redshift distribution of the identified shear-selected clusters in Table 2. We see two spikes around z of 0.3 and 0.5, which clearly indicates that this narrow 2.3 deg² field is populated by large scale structure at these redshifts; this has been mentioned already by Kubo et al. (2009). We need a significantly wider FOV to overcome the local variance and to make cosmological arguments from the redshift distribution of clusters.

The observed area that overlaps with DLS amounts to 2.3 deg², in which we found eight significant peaks whose S/N exceeds 4.5. Even if we drop peak ID 6 from the list, which may be substructure within the A781 main cluster, seven peaks still remain. Hu & Kravstsov (2003) estimated the cosmological variance in cluster samples and suggested that the variance exceeds the shot noise when the mass of clusters becomes less than $\sim 3\;\times \;{10}^{14}\;{M}_{\odot }$ with a weak dependence of the threshold mass on the survey volume. This is exactly the mass range that we are working on. Therefore, statistical arguments require comparison with cosmological simulations as we will see below.

Hamana et al. (2012) calculated the number of peaks on the weak lensing convergence map using a large set of gravitational lensing ray-tracing simulations which are detailed in Sato et al. (2009). Following that work, we made 1000 realizations to evaluate the sample variance. In making the mock weak lensing convergence map, we added a random galaxy shape noise to the lensing shear data. The rms value of the random galaxy shape noise was set so that the observed galaxy number density and rms of galaxy ellipticities are recovered. We adopted a fixed source redshift of ${z}_{s}=1.0$. Le Fèvre et al. (2013) estimated the redshift distribution of magnitude-limited samples taken from the VIMOS VLT Deep Survey, and reported the mean redshift of $\langle z\rangle =0.92$ for a sample of $17.5\leqslant {I}_{{AB}}\lt 24$ and $\langle z\rangle =1.15$ for a sample of $17.5\leqslant {I}_{{AB}}\lt 24.75$. Therefore, it would be appropriate to set ${z}_{s}=1$ for our galaxy sample of ${i}_{{AB}}\;\lt \;24.5$.

The expected peak count of S/N > 4.5 is 0.61 on 2.3 deg². In Figure 10, we show the probability distribution of the number of peaks under a given S/N threshold on a 2.3 deg² wide field. As is shown by square symbols, the maximum number of peaks reached in the realization is four (9/1000) when the S/N = 4.5. This means that we have practically no chance to have seven or eight significant peaks on a 2.3 deg² field. Is this a challenge to the current CDM-based cosmology?

Zoom In Zoom Out Reset image size

We note that the sensitivity of the number of peaks to the S/N value is quite high, reflecting the steepness of the mass function at the high mass end. So we experimentally lower the S/N down to 4 and examine the statistics. The mean number of peaks is 1.6 and the maximum number of peaks is eight in two realizations out of 1000 (0.2%, see triangles in Figure 10). This still does not reconcile the gap between the observation and the prediction.

In the meantime, Hamana et al. (2012) had adopted a cosmological simulation that used a third-year WMAP result (Spergel et al. 2007). WMAP3 is known to have yielded a relatively low ${\sigma }_{8}$ of 0.76. If we adopt the recent Planck result (Ade et al. 2015), ${\sigma }_{8}=0.83$, the expected cluster count becomes a factor of 5.26 higher than WMAP3. This relaxes the tension dramatically and now what we observed is not extremely unlikely; the chance of obtaining more than eight peaks of S/N > 4.5 is 3.7%. One thing that we could suggest here is that our peak count strongly favors the recent Planck result.

We now examine the ratio of stellar mass to halo mass, ${f}_{s}={M}_{s}/{M}_{\mathrm{halo}}$, of our samples. In the mass range that we probe, f_s is reported to decrease as the halo mass increases, suggesting that star formation is less efficient in larger halos. This can be mostly explained by the inefficiency of the cooling process inside larger halos. Gonzalez et al. (2013) presented one of the most recent results based on new halo mass estimates from XMM-Newton X-ray data. They claimed that the stellar baryon mass well compensates the shortage in the baryon budget and that the sum of the stellar baryons and the baryons in the form of gas almost reaches the universal value estimated by WMAP and Planck. They also made a comparison of different observational results (Lin et al. 2003, 2012; Gonzalez et al. 2007; Andreon 2010 ; Leauthaud et al. 2012). They found that the stellar mass fractions reported in other works are generally lower than Gonzalez et al. (2013). However, they claimed that if the mass in the intra-cluster light (ICL) is considered, the discrepancy among the observations is minimized.

Figure 11 shows the f_s versus the halo mass of our samples (circles). The mass is estimated by weak lensing as explained previously and the mass range of the samples is wider than that of previous small samples. The stellar mass M_s shown in Table 2 is the total stellar mass integrated by convolving spatial filter as shown in Figure 2 of Oguri (2014). In order to estimate the stellar mass fraction accurately, we convert this stellar mass to the stellar mass within ${r}_{{500}_{c}}$ as follows. We assume that the stellar mass density profile follows an NFW profile with the concentration parameter as a function of the halo mass of Duffy et al. (2008). For each halo with the mass ${M}_{{500}_{c}}$, we derive a conversion factor from M_s in Table 2 to the stellar mass within ${r}_{{500}_{c}}$ by convolving the projected NFW profile with the spatial filter of Oguri (2014) and estimating the ratio of the total mass with the spatial filter to ${M}_{{500}_{c}}$. We find that the conversion factor is ∼1 for massive halos with ${M}_{{500}_{c}}\sim 5\;\times \;{10}^{14\;}{M}_{\odot }$, and ∼0.5 for less massive halos with ${M}_{{500}_{c}}\sim 5\;\times \;{10}^{13\;}{M}_{\odot }$. We note that this correction also takes account of the 3D deprojection, i.e., it properly removes member galaxies outside ${r}_{{500}_{c}}$ projected along the line of sight. Thus our result should be compared with the so-called "3D" stellar mass result in Gonzalez et al. (2013) that incorporates the deprojection. The error in ${M}_{\mathrm{halo}}$ estimate is roughly 30%. We also expect a large scatter of ∼30% in M_s for a given halo mass, largely originating from the Poisson noise in the numbers of cluster members and background galaxies. Therefore, we expect a level of error as large as 40% in estimates of the individual f_s. A typical error bar is presented on the leftmost data point in Figure 11.

Zoom In Zoom Out Reset image size

Compared with the results of Gonzalez et al. (2013; triangles and the solid line in Figure 11), f_s of this work is lower on average over our mass range. We did not attempt to include the contribution from the ICL, but according to Gonzalez et al. (2013) the ICL contribution is ∼25% (dotted line in the figure), which cannot totally explain the discrepancy. Our result agrees better with lower f_s results estimated using the Halo Occupation Distribution (HOD) method in Leauthaud et al. (2012) and favors the argument that the missing baryon problem has not yet been resolved in this mass range. We further note that Gonzalez et al. (2013) assumed a stellar mass to luminosity ratio that is on average slightly lower than the Salpeter initial mass function adopted in this paper. Our stellar mass estimates are therefore larger than their estimates by $\sim 10\%$, which makes the discrepancy of the results larger by this factor.

The rate of decrease of f_s with increasing halo mass is an interesting observable because it reflects how the clusters are formed. Theoretical predictions and the recent N-body simulations coupled with semi-analytic models disfavor the steep slope in the framework of the ΛCDM cosmological model. This is because the hierarchical clustering predicts that low-mass halos with a certain f_s would be assembled together and become a larger halo with a comparable f_s. Balogh & McGee (2010) suggested that the log–log slope of ∼–0.3 would be the upper limit to be consistent with their simulations in the ΛCDM cosmology. The slopes that have been observed and were compiled in Gonzalez et al. (2013) range from –0.3 to –0.6, and the slope of –0.45 is presented by the authors' data, which is slightly steeper than the theoretical preference. In our case, the slope is slightly shallower than the data of Gonzalez et al. (2013): –0.32 when we exclude A781 main (long-dashed line) and very shallow –0.05 (short-dashed line) when we consider all the clusters. It is, however, difficult to make really qualitative arguments here due to the large error for the limited number of clusters in this work. More data are certainly called for and the on-going HSC legacy survey will provide ideal sources for future studies.