Subaru Weak-Lensing Survey of Dark Matter Subhalos in the Coma Cluster : Subhalo Mass Function and Statistical Properties **affiliation: Based on data collected from the Subaru Telescope and obtained from SMOKA, operated by the Astronomy Data Center, National Astronomical Observatory of Japan
We present a 4 deg weak gravitational lensing survey of subhalos in the very nearby Coma cluster using the Subaru/Suprime-Cam. The large apparent size of cluster subhalos allows us to measure the mass of 32 subhalos detected in a model-independent manner, down to the order of of the virial mass of the cluster. Weak-lensing mass measurements of these shear-selected subhalos enable us to investigate subhalo properties and the correlation between subhalo masses and galaxy luminosities for the first time. The mean distortion profiles stacked over subhalos show a sharply truncated feature which is well-fitted by a Navarro-Frenk-White (NFW) mass model with the truncation radius, as expected due to tidal destruction by the main cluster. We also found that subhalo masses, truncation radii, and mass-to-light ratios decrease toward the cluster center. The subhalo mass function, , in the range of 2 orders of magnitude in mass, is well described by a single power law or a Schechter function. Best-fit power indices of for the former model and for the latter, are in remarkable agreement with slopes of predicted by the cold dark matter paradigm. The tangential distortion signals in the radial range of from the cluster center show a complex structure which is well described by a composition of three mass components of subhalos, the NFW mass distribution as a smooth component of the main cluster, and a lensing model from a large scale structure behind the cluster. Although the lensing signals are 1 order of magnitude lower than those for clusters at , the total signal-to-noise ratio, S/N, is comparable to, or higher, because the enormous number of background source galaxies compensates for the low lensing efficiency of the nearby cluster.
Subject headings:galaxies: clusters: individual: (Coma Cluster A1656), - gravitational lensing: weak - X-rays: galaxies: clusters
with one command per each
The cold dark matter (CDM) concordance cosmology has had considerable success in explaining various observations on a large scale, such as the cosmic microwave background (Komatsu et al., 2011; Hinshaw et al., 2013). It provides initial conditions for the hierarchical structure formation involved in the mass assembly histories of halos, for high-resolution -body simulations and analytical models. In hierarchical clustering, less massive halos are accreted into more massive halos, which are then subsequently eroded by effects combined with tidal stripping and dynamical friction of the host halo, eventually becoming a smooth component. Since galaxy clusters are the most massive virialized objects in the universe, the central regions of subhalos have survived under the over-density field until the recent epoch, and constitute their population. Numerical simulations of, and analytic approaches to CDM predict that subhalo mass functions at the intermediate and low mass scales follow a power law, with slopes of (e.g., Taylor & Babul, 2004, 2005a, 2005b; Oguri & Lee, 2004; van den Bosch et al., 2005; Diemand et al., 2004; De Lucia et al., 2004; Gao et al., 2004b; Shaw et al., 2006; Angulo et al., 2009; Giocoli et al., 2010; Klypin et al., 2011; Gao et al., 2012; Wu et al., 2013).
Observations of cluster subhalo properties, such as mass function and spatial distribution, provide us with a deeper understanding of the mass assembly history and are the most stringent test of CDM predictions on scales of less than several Mpc. A characteristic feature of the subhalo mass function is also critically important to constrain the nature of dark matter, because it depends on the particle mass of dark matter. Furthermore, a study of the correlation between galaxy properties and subhalo masses, incorporating different data-sets, sheds important insight on the physics of galaxy evolution associated with dark matter. Thus, it is of paramount importance to measure the mass function directly from observations without assuming a relationship between dark matter and luminous matter and the dynamical state of the system. It is difficult, though, to infer the masses of subhalos from visible matter, such as galaxies, because assumptions about the mass distribution extending beyond galaxies and dynamical state of the galaxies are required. In this situation, weak gravitational lensing analysis plays an important role. Weak lensing analysis measures a coherent distortion pattern of background galaxy images caused by the gravitational field of the system and thus avoids any of the assumptions mentioned above (e.g., Bartelmann & Schneider, 2001). However, the weak-lensing signal is obtained by averaging over a certain number of background galaxies, and thus, only the mass information over a scale of several arcminutes is obtained. Previous weak lensing studies or joint strong- and weak-lensing studies (Broadhurst et al., 2005; Okabe et al., 2010b, 2011, 2013; Oguri et al., 2010, 2012; Umetsu & Broadhurst, 2008; Umetsu et al., 2010, 2011; Applegate et al., 2012; Hoekstra et al., 2012) mainly focused on clusters at redshift higher than approximately , because of good lensing efficiency and high-quality imaging data obtained using wide-field cameras mounted on ground-based telescopes, such as the Subaru Prime Focus Camera (Suprime-Cam; Miyazaki et al., 2002) on the 8.2m Subaru Telescope. However, it is very difficult to detect subhalos using weak lensing analysis of clusters at with masses on the order of , because the apparent truncation size ( or less) of these subhalos is too low to be detected in the lensing signal. Stacking lensing studies for member galaxies help to overcome this disadvantage (Natarajan & Springel, 2004; Natarajan et al., 2007, 2009; Limousin et al., 2005, 2007). It computes lensing distortion signals centered around member galaxies, and, thus, is independent of lensing selections, which increases an signal-to-noise ratio (S/N) in lensing signals due to the large sample size. Hence, the stacking analysis enables us to measure a mean mass of subhalos associated with member galaxies. However, a correlation between the luminosity and mass of the subhalos must be assumed to derive a subhalo mass function and/or conduct statistical studies of subhalos.
Weak-lensing studies of very nearby clusters () overcome the problems described above. In contrast to weak-lensing studies of clusters at , there are three significant advantages in the analysis of dark matter subhalos. First, a large apparent size enables us to easily resolve less massive subhalos inside the clusters. Second, subhalos are sufficiently separated from the main cluster center and other subhalos to ignore their lensing contamination in subhalo mass measurements. Third, a large angular scale provides a correspondingly large number of background galaxies, which leads to low statistical errors and compensates for low lensing efficiency to achieve a high S/N. This last advantage also plays an important role in cluster mass measurements. For example, the Coma cluster is at redshift , with a large apparent size times larger than that of clusters at . We thus use the area of square minutes or more in weak-lensing mass measurements of subhalos with masses greater than . Indeed, Okabe et al. (2010a) has demonstrated the power of weak-lensing analysis of the Coma cluster and discovered less massive subhalos.
Here we report the results of a 4.1 deg weak gravitational lensing survey of subhalos in the Coma cluster by 18 pointing observations ( and bands) using the Subaru/Suprime-Cam to directly measure subhalo masses and their mass function. This paper is a continuation of our previous work (Okabe et al., 2010a), which used archival Subaru/Suprime-Cam data with the band. The archival data covers the central and the south-west regions (two pointings; see also Figure 1), with a total area of deg. Our new data significantly improves the quality of the weak-lensing analysis. First, the data covers an area to the outskirts of the cluster, which enables us to study the radial dependence of subhalo properties. Area fractions, for the previous and new data, respectively, account for and within inside of which the mean interior density is times the critical mass density at the cluster redshift. Second, the exposure time is deeper than the effective one used in the previous weak-lensing analysis ( minutes;Table 1), which increases the number of background galaxies and thus suppresses the noise for the intrinsic ellipticity. Third, we used two filters to secure the background galaxies, avoiding contamination of unlensed member/foreground galaxies in our shear catalog. Therefore, this new data enables us to conduct a systematic survey for cluster subhalos, for the first time. We describe the details of data analysis in Section 2, including shape measurements, background and member selections and modeling of background lensing signals. In Section 3, we define the subhalos from lensing signals, measure model-independent projected mass, conduct stacked lensing analyses and evaluate systematic errors including the purity of the subhalo sample. We also present a galaxy-galaxy lensing study for luminous member galaxies in Section 4, which is complementary to and independent of the analyses in Section 3. We describe measurement of the main cluster mass in Section 5. Finally, we discuss the subhalo mass function, subhalo properties, and future studies in Section 6. The conclusions are stated in Section 7. Throughout this paper, we use the cosmology of , and . One arcmin corresponds to .
2. Data Analysis
2.1. Survey Observation and Image Processing
We observed the Coma cluster using the Suprime-cam (Miyazaki et al., 2002) at the Subaru 8.2-m telescope, in and bands, in 2011 March and April. The band data is used for the wide-field weak lensing analysis, and combined with band data to minimize contamination of the member and foreground galaxies in the shear catalog. The survey is covered by a mosaic of 18 pointings, specifically, coma10 … coma43, as shown in Figure 1. Each pointing overlaps by 2 arcmin. The total survey area is . A maximum projected radius from the brightest cluster galaxy, NGC 4874, reached which is comparable to the cluster virial radius (Section 5). The band data of coma30 was recollected due to the low number of background galaxies. The typical exposure times for and bands are and minutes (Table 1). We also used two imaging datasets retrieved from Subaru archival data (SMOKA111http://smoka.nao.ac.jp/index.jsp).
We used the standard pipeline reduction software for the Suprime-Cam, SDFRED (Yagi et al., 2002; Ouchi et al., 2004) modified for the new CCD, for flat-fielding, instrumental distortion correction, differential refraction, point-spread-function (PSF) matching, sky subtraction and stacking. The seeing for each pointing is shown in Table 1. An astrometric calibration was performed using point sources from the Two Micron All Sky Survey catalog (Skrutskie et al., 2006). The typical residual values are no larger than the CCD pixel size. Photometric calibration was carried out by fitting point sources detected in each dataset with stars from Sloan Digital Sky Survey (SDSS) DR8 photometry (Eisenstein et al., 2011), taking into account the difference between their sensitivities. The archival data obtained using the previous CCDs was reduced by the same procedure using the SDFRED for the previous CCDs.
Note. – aafootnotemark: : Dataset bbfootnotemark: : Exposure times in and bands, respectively. ccfootnotemark: : The seeing FWHM in unit of arcseconds, for band. ddfootnotemark: : The median stellar half-light radius in unit of arcseconds, for band. eefootnotemark: band data taken in 2011 March 1st. fffootnotemark: band data taken in 2011 March 30th. ggfootnotemark: Data retrieved from SMOKA.
2.2. Weak Lensing Distortion Analysis
The weak lensing measurements follow Kaiser et al. (1995), referred to as the KSB+ method, which uses the IMCAT package with some modifications, similar to Umetsu et al. (2010); Oguri et al. (2012); Okabe et al. (2013). Image ellipticity is measured from the weighted quadrupole moments of the surface brightness of objects detected in the band imaging data (Table 1). The anisotropic PSF correction is conducted in the same manner as Okabe et al. (2010a, b, 2011, 2013). We select bright unsaturated stars in the half-light radius, , and magnitude plane to estimate the stellar anisotropy kernel, , where is the smear polarizability matrix, and is the image ellipticity. Quantities with an asterisk denote those for stellar objects. Following the KSB method, PSF anisotropy is corrected with the equation
We estimate at each galaxy position, , using a fitting function of second-order bi-polynomials of the vector with iterative -clipping rejection. The data region is then divided into several rectangular blocks based on the typical coherent scale of the measured PSF anisotropy pattern. A number of tests were performed to assess the anisotropic PSF correction (see details in Appendix A). To estimate systematic residuals caused by imperfect PSF correction, we computed an auto-correlation function for the stellar ellipticities and a cross-correlation function for the ellipticities of galaxies and stars, before and after the correction, respectively. Although the auto correlation and the cross-correlation functions for raw ellipticities before the correction are highly corrected to the order of , the residual/corrected ellipticities show no correlation, which supports the accuracy of the anisotropic PSF correction. x
Next, the isotropic smearing effect of galaxy images is corrected to estimate the reduced distortion signal, ,
where is the pre-seeing shear polarizability tensor. The measurement of is very noisy for individual faint galaxies because of its nonlinearity (Bartelmann & Schneider, 2001), which may result in a systematic bias in weak-lensing distortion measurements. We therefore calibrate using the following procedures, in a similar way as Umetsu et al. (2010) and Oguri et al. (2012). The polarizability tensor is first computed as a scalar polarizability, . We then compute a median for in , with an adaptive grid to assemble as uniformly as possible. Here, is the Gaussian smoothing radius used in the KSB method. A sample of galaxies satisfies the following conditions to suppress the noise: a detection significance level of , a size condition of and and a positive raw . Here, () and () are the median (rms dispersion) of half-light radii and Gaussian smoothing radii for the stars selected above. We interpolate the polarizability tensor for individual galaxies as a function of . A similar interpolation for the absolute value of the ellipticity, , is also applied. We use galaxies for the shear catalog with and the same size cut as in the calibration. An rms error of the shear estimate, , is computed from 50 neighbors in the magnitude- plane. We also assign the weight function for individual objects.
where is the softening constant variance representing the scatter due to the intrinsic ellipticity of the galaxies (e.g., Hoekstra et al., 2000; Hamana et al., 2003; Okabe et al., 2010b; Umetsu et al., 2010; Oguri et al., 2012). We choose , which is a typical value of the mean rms over the background sample, In the limit of , individual galaxies are uniformly weighted. On the other hand, noisy objects, such as fainter objects, are less weighted.
To check shear calibration, we use a number of realistic images for which the field-of-view is comparable to the Subaru/Suprime-cam (kindly provided by M. Oguri). The mock images are generated with different seeing sizes (0.5â1.1 arcsec) and the Moffat profile with power slopes , using GLAFIC software (Oguri, 2010), as described in Oguri et al. (2012). We found that a multiplicative calibration bias, and an additive residual shear offset, in Heymans et al. (2006) and (Massey et al., 2007) are and , respectively, for our typical seeing .
We then combine the shear catalog constructed from individual images. For the overlapping regions, since the same galaxies are detected in different images, we estimate weighted averages of their position and shear with . We compared reduced shear for overlapping galaxies and confirmed that the deviation, , is negligible. Using this approach, the number of background galaxies is .
2.3. Photometry and Background Selection
A secure selection of background galaxies in the color-magnitude plane was used because contamination by unlensed member or foreground galaxies in the shear catalog dilutes the weak-lensing signals, leading to an underestimation of the gravitational lensing mass, mainly for the central regions (Broadhurst et al., 2005; Okabe et al., 2010b, 2013).
Photometric catalogs were constructed from the mosaic images using SExtractor (Bertin & Arnouts, 1996). The SExtractor parameters are optimized for faint galaxies for shape measurements. We compute the total magnitude for each object in the AB-magnitude system using the MAG_AUTO parameter and color using the MAG_APER parameter. For the color measurements, we degraded the seeing to the worst image. The aperture diameter for the MAG_APER parameter is times the seeing FWHM. The overlapping galaxies serve as a monitor of the offset in the magnitude. We introduced an additional parameter in each data field to describe the offset using two magnitudes and calibrated them simultaneously fitting bright objects with magnitudes less than mag. The measurement scatter for faint galaxies ( mag) is typically less than mag. The magnitudes and colors for objects are estimated using weighting measurement errors. We then match the shear and SExtractor catalogs.
The red-sequence of member galaxies is fitted to a linear function, using luminous galaxies ( mag). We then define the background galaxies with colors redder than the red-sequence in the magnitude range of mag mag (Figure 2). The number of background galaxies is reduced to after the color cut but remains times higher than those of clusters at , obtained by previous studies using two path-band filters (Okabe et al., 2010b). The number density, , is also from two to eight times higher than both those for clusters at , and for our previous analysis of the Coma cluster (Okabe et al., 2010a, ). Thus, we can use a correspondingly large number of background source galaxies for nearby cluster weak-lensing analysis, for the following two reasons. First, since the colors of red-sequence galaxies in clusters becomes more blue with decreasing redshifts, the number of galaxies, with colors are redder than those of member galaxies, increases. Second, the large area encompassed by the nearby cluster increases the number of background galaxies. Even if member galaxies are contained in the background catalog, the dilution effect in lensing signals could be ignored because the ratio of thousands of member galaxies to the millions of background galaxies is negligible. Thus, weak-lensing analysis of a nearby cluster has a great advantage to compensate for low-lensing efficiency.
2.4. Mean Lensing Depth
Since the redshifts of individual galaxies in the shear catalog are not available, we estimated the mean source redshift using a statistical approach. The lensing signal depends on the source redshifts through the distance ratio. As a reference, we used the COSMOS photometric redshift catalog (Ilbert et al., 2009) estimated by combining 30 broad, intermediate and narrow bands. Because the band is not available in the COSMOS catalog, and we converted magnitudes from to bands, based on the filter sensitivities of the Suprime-Cam. The probability function of redshift, , for our background galaxies selected by the color-magnitude plane (Section 2.3) is computed by matching with the COSMOS photometric redshift, with a statistical weight of . The mean distance ratio is given by
where and are the angular diameter distance to the sources and between the cluster (lens) and the sources, respectively. We obtain . The mean source redshift, , is slightly lower than that () for clusters at (Okabe et al., 2013), because we include many background galaxies at lower redshifts. The mass estimate for nearby systems do not strongly depend on the redshift distribution of background sources.
2.5. Luminous Member Galaxies
We defined luminous member galaxies identified in spectroscopic observations in order to compare the mass properties. Luminous galaxies with a magnitude brighter than , were retrieved from SDSS DR8 (Eisenstein et al., 2011), in a deg region centered on NGC4874. Furthermore, member galaxies were selected within the redshift range of and , where is the velocity of light. To complete the catalog of member galaxies on the bright end, we also checked the redshifts of galaxies in NED222http://ned.ipac.caltech.edu/. If they satisfied the above conditions, they were added to the catalog. Luminosities of individual galaxies are estimated from apparent magnitudes using the k-correction for early-type galaxies, assuming a single redshift of .
2.6. Model of LSS Lensing
Weak-lensing mass measurements for clusters at low redshift suffer from lensing signals of background galaxies between the cluster and the source redshifts, referred to as uncorrelated LSS lensing. The three-dimensional, inhomogeneous mass distribution causes a locally strong shear pattern, which potentially gives biases in detection and mass measurements of localized objects, such as subhalos. We therefore quantified the uncorrelated LSS lensing effect on each galaxy in the shear catalog, using the luminosity and photometric redshift retrieved from SDSS DR8 (Eisenstein et al., 2011), following Okabe et al. (2010a). Galaxies are selected with magnitudes mag and photometric redshifts between the cluster, , and the source redshift, where is the photometric redshift of each galaxy. Galaxies spectroscopically identified as member galaxies (Section 2.5) were excluded. Masses of individual galaxies are estimated using galaxy-galaxy lensing results from SDSS data Guzik & Seljak (2002). Using the mass-to-light ratio in each band () derived by stacked lensing analysis of galaxies, the luminosity is converted into the mass. The masses estimated with different bands are used to cross-check and calibrate systematic errors in the mass-luminosity scaling relation utilized here. Since uncorrelated LSS lensing is obtained by integrating the effect of light deflections due to galaxies at different redshifts along the line-of-sight, the best-fit scaling relations in mass estimates for individual objects were used in order to quantify an average LSS lensing effect. The interior mass structure of each halo is assumed to be a universal mass profile found in numerical simulations, referred to as NavarroâFrenkâWhite (NFW; Navarro et al., 1996, 1997). The NFW mass model is described by two parameters, the mass and concentration (see details in Appendix D). It is well known that there is a correlation between mass and halo concentration (Bullock et al., 2001) predicted by the hierarchical structure formation scenario. We use the mass-concentration relation obtained from recent numerical simulations based on WMAP5 cosmology parameters (Duffy et al., 2008) to describe the internal structure. The tangential distortion signals of individual galaxies are computed on all source galaxies. We found that the shear estimated in the band is consistent with that in the band, but the estimates in the bands are systematically different, as found in Okabe et al. (2010a). The LSS lensing model based on the galaxy-galaxy lensing result is therefore defined with . If groups or other clusters exist behind the cluster, this model would fail to incorporate those effects. Since this would bias the mass measurement of the main cluster, this is considered separately in Sections 3.4.1 and 5. The LSS lensing model allows us to statistically estimate the lensing signals of real background structure from the observing data and the LSS bias in the mass measurement. We thus use the LSS lensing model for the main analysis of this paper.
We also conduct the cluster and subhalo mass measurements taking into account the error covariance matrix of uncorrelated large-scale structure along the line-of-sight, (e.g., Schneider et al., 1998; Hoekstra, 2003; Umetsu et al., 2011; Oguri et al., 2010; Oguri & Takada, 2011; Okabe et al., 2013), instead of the above LSS lensing model. The LSS error covariance matrix is estimated from the weak-lensing power spectrum (e.g., Schneider et al., 1998; Hoekstra, 2003) with WMAP7 cosmology (Komatsu et al., 2011). Since we fully take into account both the LSS error matrix and the statistical noise caused by the intrinsic shapes of the galaxies and the noise in the shape measurement, this approach is complementary to the LSS lensing model. However, the statistical error is dominated in subhalo mass measurements (see Section 3.4.2), and it is difficult to identify real background structure using the error matrix.
3. Weak-Lensing Analysis for Subhalos
3.1. Projected Distributions of Mass and Baryons
We first make maps of the lensing convergence field (), luminosity () and number density () of member galaxies and the model of the LSS lensing signal (). In order to identify subhalos in a model-independent way, the projected mass distribution is reconstructed following Kaiser & Squires (1993) with a Gaussian smoothing kernel. The details of map-making are explained in Appendix B. We adopt various smoothing scales in the range of arcmin, stepped by for arcmin and for arcmin, to optimize the detection of subhalos with various mass properties. The definitions of subhalos are described in the next subsection. We present here the correlation between mass and luminous matter on the projected distribution.
Figure 3 shows the significance map, , defined by , with a smoothing scale of (), where the reconstruction error, , is calculated over local background galaxies (see Appendix B) where a typical value in this smoothing scale is . The LSS lensing model was not taken into account. The mass distribution in the central region, in which two cD galaxies NGC 4874 (,)=() and NGC 4889 () exist, is elongated in the east and west directions. Clumpy structures are found everywhere, but anisotropically distributed. In particular, the projected distribution of clumpy structures is concentrated southwest of NGC 4874. Some clumpy structures are associated with background groups in the literature (letters in Figure 3 and Table 2).
The left and middle panels of Figure 4 show maps of luminosity and the number density of member galaxies, overlaid with the contours of the projected mass distribution. The mass and galaxy distribution are clearly correlated with each other. The right panel of Figure 4 is the convergence map of the LSS lensing model. The S/Ns of the LSS lensing map are at most , and thus, the LSS lensing effect accounts for a small fraction of the observed signal. However, the LSS model fails to describe the lensing signals around some groups behind the cluster (Table 2). The reason is likely that the estimation of the LSS lensing effect is based on galaxy-galaxy lensing which fails to take group-scale or cluster-scale structures into account. This is further elaborated in the Section 3.4.1.
|A||MaxBCG J195.08820+26.78870||Koester et al. (2007)|
|B||GMBCG J195.47315+26.95810||Hao et al. (2010)|
|C||MaxBCG J195.47907+27.16429||Koester et al. (2007)|
|D||GMBCG J195.34791+29.07201||Hao et al. (2010)|
|E||MaxBCG J195.34617+29.18616||Koester et al. (2007)|
|F||NSC J125939+290715||Gal et al. (2003)|
|G||GMBCG J193.96542+28.51557||Hao et al. (2010)|
|H||MaxBCG J193.92901+28.76123||Koester et al. (2007)|
|I||SDSSCGB 06685||McConnachie et al. (2009)|
|J||WHL J125535.3+273104||Wen et al. (2009)|
Note. – aafootnotemark: :The identification of background systems in Figure 3. bbfootnotemark: :Photometric redshifts.
To quantify the correlations shown in the maps, we compute the pixel-to-pixel coefficients between the mass maps () and the luminosity () and density () maps for member galaxies. The resultant coefficients for both and change from () to () with a decrease in spatial resolution. Here, the errors are estimated by bootstrap re-sampling with 200 realizations of maps, describing that noise peaks are accidentally correlated with smoothed luminous distributions. In short, the correlation between mass and member galaxy distributions is at the level of -. We also computed the coefficients between the mass map and the LSS lensing map (), and found high correlation, () for and () for , respectively. The significance level is higher with an increase in resolution, indicating that the LSS lensing signal caused by small background objects creates a local shear pattern.
As shown in Figure 5, the X-ray surface brightness distribution of ROSAT X-ray satellite shows an elongated X-ray distribution in the central region and an excess X-ray flux associated with the NGC4839 group in the southwest direction (Briel et al., 1992; White et al., 1993; Neumann et al., 2001). Mass contours are overlaid with a smoothing scale of FWHM to compare with the diffuse emission detected with a large PSF of the ROSAT. Although the diffuse X-ray emission from the NGC4839 group is associated with a clumpy mass structure, all mass structures are not necessarily correlated with X-ray features. This point is further discussed in Section 6.6.
3.2. Selection and Mass Measurements of Subhalos
We explore, in a model-independent way, subhalo candidates by finding peaks in the mass maps reconstructed using several smoothing scales. As described in Sec 3.1, maps of the observed lensing signals are correlated with those of the LSS lensing model (Figure 4). In order to securely identify cluster subhalos and accurately measure their masses, it is crucial to minimize the contamination by the LSS lensing effect. We therefore calibrate the reduced shear with an approximate form of to eliminate the LSS lensing effect along the line-of-sight as much as possible. Mass reconstructions are then repeated using the calibrated shear catalog.
Subhalo candidates with peaks above a threshold in the mass maps are selected. The mass maps are represented as the convolution of the lensing distortion pattern of a cluster mass distribution with smoothing kernels. Therefore, the Gaussian smoothing scales used for the mass reconstruction vary from to in order to optimize for the detection of subhalos with various mass properties. Here, the pixelized field changes slightly using the reconstruction kernels, similar to top-hat or wavelet filters.
We use a significance level, , for the selection of subhalo candidates, where and are the dimensionless surface mass density and the reconstruction error, respectively. Since the variance and skewness of the histogram in the pixels depend on the smoothing scale, we identify subhalo candidates above a threshold set at three times the standard deviation. The threshold of significance in the highest resolution corresponds to . We first identify subhalo candidates at various smoothing scales. Then, two peaks appearing between two different smoothing scales are matched with the condition , where is the distance between the two peaks which appeared in different scales, and is the full width and half the maximum of the larger smoothing scale. This process results in 49 subhalo candidates. We note that Okabe et al. (2010a) used a mass map with single smoothing scale (FWHM=) and applied the lower threshold. Therefore, two of the seven subhalo candidates in the previous paper (Okabe et al., 2010a, numbers 6 and 8) are below a more conservative threshold of this analysis.
Since we minimized the LSS lensing contribution by applying the galaxy-galaxy lensing model, eight known background objects (Table 2) are below the thresholds selected. However, the model does not perfectly describe the full LSS lensing effect. Three other peaks associated with the known background objects (Table 2) are detected with the above conditions. One is the background object “I” and two peaks are around the object “F” (see Figure 3). These objects are likely to be groups because the lensing signals are stronger than what is expected from the luminosity of a single galaxy. Furthermore, there is a possibility that background groups are accidentally superimposed with cluster subhalos, giving a systematic bias on mass estimates of subhalos. This point is discussed in Section 3.4.1.
Next, we measure the model-independent projected masses (Clowe et al., 2000, see also Appendix C) for shear-selected subhalo candidates. This measurement has several important advantages. First, a large number of background galaxies are available, because a projected mass within a circular aperture radius is computed by integrating source galaxies outside the radius. The measured projected mass is a cumulative function of radius. Thus, this approach suppresses the random noise relevant to the intrinsic ellipticity, compared to a tangential distortion profile, which averages the tangential component of all background galaxies residing in radial bins. Second, since the measurement subtracts the background mass density surrounding subhalos, the contribution of the main cluster mass distribution to subhalo masses is excluded. Third, the mass density of subhalos is expected to be close to zero outside of the tidal radius, and the measured aperture mass corresponds to the subhalo mass itself. If the mass density profile follows the universal NFW profile (Navarro et al., 1996, 1997) without any truncation radii, the aperture mass is higher than the spherical one (Okabe et al., 2010b). As expected from tidal destruction, the radial profile of the projected mass is saturated outside the truncation radii, . We measure projected masses for all the candidates. Since the smoothing kernel for the mass reconstructions gives rise to centroid uncertainties of the candidates, we determine the central position by choosing maximal lensing signals within a box where the center is aligned with the map peak position. For accurate mass measurements of subhalos with a variety of sizes, it is important to explore truncation radii where the projected mass profile is saturated. We systematically compute projected mass profiles by changing the background annulus and then statistically determining the truncation radii. Here, the inner radius changes from to in steps of and the width is fixed at . The projected mass is computed from saturated values, taking into account the error covariance matrix. The measurement method is detailed in Appendix C. The same analysis was repeated for different background widths which showed that the result does not significantly change. Mass measurements used a considerably large number of source galaxies (). The number is comparable or less than that for main clusters at (e.g., Okabe et al., 2010b) for which the background number densities are . Less massive subhalos which are detected inside more massive ones should be excluded in order to avoid double-counting these subhalos. We count the th subhalo using two conditions of the radius and the subhalo mass (). The number of candidates is then reduced from 49 to 39 using this procedure. As mentioned above, the LSS model fails to fully explain the lensing signals of background systems, especially on group scales. Furthermore, since there is a possibility to detect mass structures behind the cluster, we conservatively select the candidates hosting spectroscopically identified member galaxies within their truncation radii as the cluster subhalos. Having applied these limitations, 32 peaks are identified as dark matter subhalos. Three candidates are associated with the background systems (Table 2). Four candidates have no optical counter: they are located around in the south-east direction and the north-west direction, respectively.
These 32 subhalos are labeled by integers, in the order of right ascension. The resulting subhalo masses, , range from to (Table 3). As shown in Figure 6, the radial profiles of the projected mass clearly show saturation at some outer radii. The subhalos are widely distributed from the northeast to the southwest in the sky (Figure 3). Interestingly, the direction connecting between the Coma cluster and A1367 which are parts of the Coma supercluster (Gregory & Thompson, 1978) agrees roughly with the subhalo distributions. Several massive subhalos are associated with well-known, spectroscopically identified groups in the cluster (e.g., Mellier et al., 1988; Adami et al., 2005). Galaxies or groups associated with subhalos are summarized with references in Table 3. The cD galaxies, NGC4874 and NGC4889, are associated with subhalos “21” and “24”, respectively. The mean mass ratio reported in this paper compared to the previous paper for overlapping subhalos is . We also measured the projected masses for two subhalos with peaks below the threshold in this analysis. The mean mass ratio is . Since the number density of background galaxies in the previous analysis is about half of that reported in this analysis, we cannot rule out the possibility that these peaks are actually above the threshold.
|2||(194.011, 27.685)||NGC4816 Group$\sharp$$\sharp$footnotemark:|
|3||(194.197, 27.763)||SDSS J125645.42+274638.0|
|4||(194.232, 27.053)||SDSS J125647.00+270324.9|
|5||(194.298, 27.360)||2MASX J12571076+2724177|
|6||(194.355, 27.660)||G12 Group$\ddagger$$\ddagger$footnotemark:|
|7||(194.361, 28.187)||UGC08071, 2MASX J12572841+2810348|
|8||(194.372, 27.380)||2MASX J12573148+2723048|
|9||(194.381, 27.493)||NGC4839 Group$\sharp$$\sharp$footnotemark: , G4 Group$\natural$$\natural$footnotemark: , NGC4842, X-ray subhalo$\flat\flat$$\flat\flat$footnotemark:|
|10||(194.477, 28.507)||2MASX J12575392+2829594|
|11||(194.572, 27.745)||2MASX J12581922+274543|
|12||(194.579, 27.846)||SDSS J125818.20+275054.5|
|13||(194.597, 27.101)||2MASX J12581552+2705137|
|15||(194.656, 27.905)||NGC4839 Group$\sharp$$\sharp$footnotemark:|
|16||(194.659, 26.738)||SDSS J125839.93+264534.2|
|17||(194.718, 27.825)||G9 Group$\natural$$\natural$footnotemark: , SA 1656-030$\flat$$\flat$footnotemark:|
|18||(194.732, 27.759)||G8 Group$\natural$$\natural$footnotemark:|
|19||(194.790, 28.288)||SDSS J125914.99+281503.6|
|20||(194.879, 28.062)||2MASX J12593141+2802478|
|21||(194.882, 27.936)||NGC4874(cD),part of G1 Group$\natural$$\natural$footnotemark: , X-ray subhalo 2$\sharp\sharp$$\sharp\sharp$footnotemark:|
|22||(194.895, 28.511)||2MASX J12594129+2830257|
|23||(194.971, 27.837)||J194.9353+27.83393$\ddagger$$\ddagger$footnotemark: , SA 1656-054$\flat$$\flat$footnotemark: ,X-ray subhalo 3$\sharp\sharp$$\sharp\sharp$footnotemark:|
|24||(195.052, 28.005)||NGC4889(cD),part of G1 Group$\natural$$\natural$footnotemark: X-ray subhalo 1$\sharp\sharp$$\sharp\sharp$footnotemark:|
|25||(195.086, 28.542)||2MASX J13002268+2834285|
|26||(195.111, 28.654)||SDSS J130037.14+283950.9|
|27||(195.115, 28.080)||SDSS J130030.95+280630.2,part of G7 Group$\natural$$\natural$footnotemark:|
|29||(195.220, 28.010)||NGC 4908, NGC 4908 Group|
|30||(195.300, 28.558)||SDSS J130114.96+283118.3|
|31||(195.325, 27.830)||G4 Group$\natural$$\natural$footnotemark: , NGC4919|
|32$\dagger\dagger$$\dagger\dagger$footnotemark:||(195.421, 29.054)||G15 Group$\natural$$\natural$footnotemark: ,IC 4088,2MASX J13014399+2859587|
Note. – aafootnotemark: : Subhalo name bbfootnotemark: : Weak-lensing center in units of deg ccfootnotemark: : Subhalo mass in units of . ddfootnotemark: : Maximum signal-to-noise ratio appearing in the mass maps (). eefootnotemark: : Name of representative galaxies or optical groups. $\dagger$$\dagger$footnotemark: : Possibly an overlapped background structure, WHLJ125535.3+273104 (Table 2). $\dagger\dagger$$\dagger\dagger$footnotemark: : Possibly an overlapped background structure, GMBCG J195.34791+29.07201 (Table 2). $\sharp$$\sharp$footnotemark: : Mellier et al. (1988) $\natural$$\natural$footnotemark: : Adami et al. (2005) $\flat$$\flat$footnotemark: : Conselice & Gallagher (1999) $\ddagger$$\ddagger$footnotemark: : Adami et al. (2009) $\sharp\sharp$$\sharp\sharp$footnotemark: :Andrade-Santos et al. (2013) $\flat\flat$$\flat\flat$footnotemark: :Briel et al. (1992)
3.3. Stacked Lensing Analysis for Subhalos
Next, we conducted a stacked lensing analysis for the subhalo candidates, which is complementary to the projected mass measurement. The power of the stacked lensing technique is to reduce the random noise due to intrinsic ellipticities by increasing the number of source galaxies. Tangential profiles, even for small and less massive subhalos, can be computed and their average parameters can be determined with lower measurement errors.
We first divide the subhalos into three subsamples based on the model-independent projected masses of , and . The mass thresholds are chosen by a subhalo mass function which is described in Sec 6.1. The number of subhalos are 21, 8, and 3, progressing from less massive to more massive subhalos. Figure 7 shows that the tangential component is positive (top panel) and the rotated component is positive and negative in random order (bottom panel). The mean of the rotated component over the radial range is consistent with a null signal, within the error of the mean. A sharply truncated feature is found in the stacked signal of the tangential profile. Outside the breaks, the profiles are proportional to , which indicates that the mass density becomes zero. We emphasize that such a feature was not found in massive clusters (e.g., Okabe et al., 2010b, 2013) but was identified in the stacked lensing profile for subhalos in our previous paper (Okabe et al., 2010a). We did not apply any rescaling procedures to the radial bins corresponding to the lensing signals, because this mass weight scheme biases the mass estimates, as described by Okabe et al. (2013). Here, an off-centering effect (Yang et al., 2006) in the lensing signals from the main cluster mass is negligible because the mean projected distance from the cluster center is much larger than the maximum radius for the plots. The stacked lensing profiles are then fit with NFW, TNFW and TNFWProb models (Appendix D). Here, the TNFW model is a truncated NFW (Takada & Jain, 2003; Okabe et al., 2010a). The TNFW model is an extreme case of truncation models, where the mass density outside the truncation radius is zero as described in Appendix D. The TNFWProb model is the TNFW model taking into account a probability function for the truncation radius which is assumed to be Gaussian with the mean, , and the standard error . Given this function, we measure a mean subhalo mass . In the process of fitting the model, we propagate systematic errors by possible background structures around subhalos “1” and “32”, which is described in 3.4.1. As expected from the clear truncation feature, the mean tangential profiles are well fitted using the TNFW and TNFWProb models (Figure 7). The best-fit subhalo masses and truncation radii are listed in Table 4. The best-fit mass and truncation increase with increasing model-independent projected masses. If the subhalo sample was entirely from false peaks, these characteristic features could not be found. We compute the significance probability, , that the data shows as a poor fit, as the observed value of by chance. The NFW models for the lowest- and highest- mass samples are rejected with a significance level of . Thus, the NFW model is inadequate to describe the tangential shear profile with breaks. The mean ratio between the best-fit subhalo masses and the mean projected mass of the subsamples, , are for the TNFW model and for the TNFWProb model, respectively, These two models are in good agreement.
Note. – aafootnotemark: : Name of subsamples for subhalos in the stacked lensing analysis bbfootnotemark: : Number of subhalos ccfootnotemark: : Best-fit subhalo mass and truncation radius for the TNFW model ddfootnotemark: : Best-fit subhalo mass, and the average and standard error of the truncation radius for the TNFWProb model eefootnotemark: : Average luminosity for associated galaxies, estimated by weighting tangential distortions, fffootnotemark: : Signal-to-noise ratio for the tangential distortion profile ggfootnotemark: : Probabilities that the TNFW mass and truncation radius represent false subhalos are within contours for the best-fit values of observed subhalos $\dagger$$\dagger$footnotemark: : The NFW model is strongly disfavored
Next, we repeat the stacked lensing analysis for four subsamples divided by the projected cluster-centric radii (,,, and ). Since tidal destruction predicts that the truncation radii are statistically correlated with the three-dimensional radius, a stacked procedure averages out line-of-sight positions for subhalos. This provides information regarding the dependence of mean subhalos size on the cluster-centric radius. Figure 8 displays the mean tangential profiles with clear breaks. The TNFW and TNFWProb models give a better to fit the stacked profiles (Table 4). Although the NFW model fit is acceptable (), the TNFW and TNFWProb are preferred based on comparing the goodness-of-fit of each model. The mean truncation radius increases as the projected radius from the cluster center increases. The mean mass ratios are , and for the TNFW and TNFWProb models, respectively.
Previous papers (e.g., Natarajan et al., 2007, 2009; Limousin et al., 2005, 2007) estimated subhalo masses by galaxy-galaxy lensing method using a model of a pseudo-isothermal elliptical mass distribution (PIEMD) of which three-dimensional mass density is given by . Here, the core radius is at the order of 100 pc (Limousin et al., 2005) and is the truncation radius. The tangential shear for the PIEMD model (Natarajan et al., 2007) falls as in the transition region () and in the outer region (). We also tried to fit the stacked lensing signals with the PIEMD model. Here we assume that the core radius is one-hundredth of the truncation radius, because there is no data on scales of sub kpc and thus we cannot constrain it. We also assume a spherical model for the simplicity. The PIEMD model gave a poor fit because the slope of the model in the transition region () is different from the observed profile for the mass scales of our subhalos.
3.4. Systematic Errors
In this section, we assess various systematic errors on the subhalo analysis, such as a projection effect on subhalo mass measurements (Section 3.4.1), LSS error covariance matrix (Section 3.4.2), a probability of spurious peaks (Section 3.4.3), selection criteria of subhalos (Section 3.4.4) and stacking procedure (Section 3.4.5). They are critically important for further discussion of subhalo properties such as a mass function (Section 6.1; construction of Figure 13). Each systematic error would have an independent effect on the mass function. For instance, the projection effect would lead to a bias in subhalo mass measurements (the -axis of the mass function). A contamination of spurious peaks and selection criteria would change a shape of the mass function (the -axis), especially on small mass scales.
3.4.1 Projection Effect
The projection effect on lensing mass measurement of cluster subhalos, caused by background groups accidentally superimposed along the line-of-sight, is examined here. Although LSS modeling is quantified based on scaling relations between mass and luminosities, it fails to fully describe massive background structures, such as groups or clusters. This effect would lead to a bias in mass estimates, possibly changing the mass of the x-axis in the mass function (Section 6.1; Figure 13). As shown in Figure 3, possible background structures, J and D (Table 2), are located within two subhalo regions labeled “1” and “32” (Table 3), respectively. Although it is in principle very difficult to discriminate between them from the observed lensing signal, a difference between the expected mass density profiles enables us to assess a contribution from background structures in the observed lensing signal. Since interior subhalos are tidally destroyed by their parent halos, it is expected that the mass density profile outside the tidal radius sharply declines. On the other hand, virialized background groups or clusters do not show such a feature as long as there is no neighboring massive halo. Indeed, tangential distortion profiles for individual groups or clusters and stacked profiles show a clear curvature as a characteristic signature of the NFW prediction and no evidence of a truncation feature (e.g., Johnston et al., 2007; Okabe et al., 2010b, 2013; Umetsu et al., 2011; Oguri et al., 2012; Taylor et al., 2012). Thus, fitting models to tangential distortion profiles helps us to discriminate between subhalos and background objects. As mentioned above, the profile for less massive subhalos is very noisy and slightly changed by the choice of radial bins, because the number of source galaxies is small in proportion to the area surrounding less massive subhalos. We therefore concentrate on computing distortion profiles for three massive subhalos (“1”, “9” and “32”), with masses greater than .
Figure 9 displays breaks in the tangential shear profiles. The slope follows outside the break, as shown by stacked lensing analysis (Section 3.3). The off-centering effect (Yang et al., 2006) of the main cluster mass on the lensing signal is negligible because of large separation from the cluster center. We first fit the TNFW model as a model of subhalos to the tangential distortion profiles. We also tried to fit a truncated singular isothermal sphere (TSIS; Okabe et al., 2010a) model to the profile but found a poor fit for massive subhalos. The best-fit masses and truncation radii are shown in Table 5. We found that the best-fit values do not change significantly by a choice of radial bins. The solid lines for the best-fit values describe the profiles with the breaks (Figure 9) well. Next we fit the NFW model to the data and then obtain larger minimum reduced values than those for the TNFW model. The significance probabilities, , for the NFW model are , , and for subhalos “1” , “9”, and “32”, respectively. When we adopt the threshold of , the NFW model for subhalo “32” is unacceptable and for subhalos “1” and “9” are acceptable although for subhalo “1” is close to the threshold. If observed lensing signals around subhalos “1” and “32” were explained only by background objects, the profiles should be well described by the NFW model. We repeat the tangential fits using photometric redshifts and as redshifts of background objects around subhalos “1” and “32”, and obtain the virial masses for the NFW model, , and , respectively. In this mass scale, no clear truncation radius was found in the tangential profiles (e.g., Okabe et al., 2010b; Oguri et al., 2012). Thus, it implies that all lensing signals cannot be explained solely by background objects. We then fit using a combined model of the TNFW for subhalos and NFW for backgrounds, where we assume the mass and concentration relation for backgrounds (Duffy et al., 2008) and fix the truncation radius derived by fitting the TNFW model. Although measurement errors of subhalo masses become larger, we find that the best-fit subhalo masses are decreased by 30%-40%. We therefore add these differences to the second error in Table 5, as the systematic error, and propagate them into the stacked lensing analysis (Section 3.3; Table 4).
Note. – aafootnotemark: : Name of subhalos (Table 3). bbfootnotemark: : Best-fit mass, in units of . ccfootnotemark: : Best-fit truncation radii, in units of . ddfootnotemark: : Reduced chi-square for the best-fit truncated NFW (TNFW) model (d.o.f is the degrees of freedom). eefootnotemark: : Signal-to-noise ratio for the tangential distortion profile. fffootnotemark: : Reduced chi-square for the best-fit NFW model. All are higher than those of the TNFW model. In particular, the significance probability, , for the NFW model of subhalo ’32’ is less than 10%, indicating that the profile is not well fitted by the NFW model.
3.4.2 LSS Error Covariance Matrix
An alternative approach to take into account LSS lensing effect in weak-lensing mass measurements is to use the error covariance matrix of uncorrelated large-scale structure along the line-of-sight (e.g., Schneider et al., 1998; Hoekstra, 2003), instead of the LSS lensing model. Here, We estimate the error covariance matrix in the - and -th radial bin, where is a diagonal matrix of the uncertainty caused by the intrinsic shapes of the galaxies and the noise in the shape measurement, and is calculated by the weak-lensing power spectrum (e.g., Schneider et al., 1998; Hoekstra, 2003) with WMAP7 cosmology (Komatsu et al., 2011). The diagonal component of LSS error covariance matrix, , is lower than the statistical error for and comparable to those for , respectively. Thus, the statistical error is denominated in the radial range of the subhalo mass measurements. We computed the stacked tangential shear profiles from the shear catalog without LSS lensing model. As the truncation position in the tangential shear profiles does not change significantly, the best-fit truncation radii agree within with those in Table 4. As the lensing signals at the truncation radii become higher than those estimated from the shear catalog with the LSS lensing, the subhalos masses for mass and radial bins become and higher. In other words, our LSS model corrects the LSS lensing bias by . The measurement uncertainties of the subhalo mass and the truncation radius, estimated with the error covariance matrix, are consistent with those estimated from the statistical error, because the statistical error is dominated.
3.4.3 Probability of Spurious Peaks
To measure a reliable subhalo mass function, it is of critical importance to statistically rule out the possibility that the subhalo candidates are actually spurious peaks. The peak finding method always suffers from the presence of spurious peaks. It is therefore of vital importance to quantify the number and properties of spurious peaks in order to confirm the purity of real subhalos. Especially, if our subhalo catalog included spurious peaks, a shape of a subhalo mass function (Sec 6.1;Figure 13) would be changed. For this purpose, we create 200 bootstrap data-sets generated by randomly swapping reduced shear at fixed positions, repeat the map making process and then identify artificial false peaks which satisfy the same conditions except for the spectroscopic information of the galaxies. The number of spurious peaks is for each realization. Thus, we cannot completely rule out a contamination in the subhalo catalog. It is, however, difficult to quantify the purity of subhalos comparing the number of detected subhalo candidates and false peaks, because we excluded several subhalo candidates taking into account the cloud-in-cloud problem and the background groups. Stacked lensing analysis enables us to measure the mean parameters even for spurious peaks, although individual measurements of spurious peaks are very difficult due to high statistical noise. Comparing the statistical properties of spurious peaks allows us to discuss the purity of the subhalo catalog. We generated 500 bootstrap replications of stacked tangential profiles using the catalog of artificial false peaks. Here, the number of spurious peaks and radial bins are the same as those for subsamples in stacked lensing analysis of subhalos (Sec 3.3). The mean mass and truncation radius for spurious peaks are estimated by fitting with the TNFW model. We found that of profiles for spurious peaks gives a poor fit () or are ill-constrained. This indicates that the profile shapes are different from those of observed subhalos. Indeed, the stacked lensing analysis for shear-selected subhalos shows that the best-fit truncation radius depends on the projected mass and the cluster-centric radius (Figures 7 and 8). If our sample consisted entirely of spurious peaks, such a clear dependence could not be found. To make a more robust conclusion, we estimate the probability that the parameters for spurious peaks accidentally coincide with those for observed subhalos within 1 uncertainty, based on Monte Carlo re-distributions of the best-fit values with the covariance matrix of the measurement errors. The false probabilities for individual subsamples in stacked lensing analysis , , are from to (Table 4). Multiplying the number of subhalos by the false probability in each sub-sample, the expected numbers of spurious subhalos are less than unity. Therefore, we conclude that our sample of subhalos has a high degree of purity.
3.4.4 Selection Criteria
A choice of the threshold in the S/N for mass maps results in one of the main systematic errors, because we cannot completely rule out the possibility that peak heights in mass maps are accidentally above or below the threshold due to reconstruction errors of finite sampling of background galaxies. Assuming Poisson fluctuations of the reconstruction noise, the S/N changes by , where is an effective number density in the smoothed mass map with the highest resolution. We repeated the same analysis with different thresholds. The sample numbers become 24 and 49 with higher and lower thresholds, respectively. This systematic error is taken into consideration to compute a subhalo mass function in Sec 6.1.
3.4.5 Stacking Procedure
We investigate whether the stacking method gives systematic errors, because the mean tangential profile stacked over subhalos with various truncation radii would blunt the break feature. We make synthetic weak shear catalogs of subhalos using the analytic TNFW model and the intrinsic ellipticity. Here, the number of background sources is the same as that observed. The parameters of the TNFW model for individual subhalos are generated from a Gaussian distribution. The mean and standard error of subhalo masses and the truncation radii for simulated samples are shown in Table 6. We assume that the coefficient between the subhalo mass and the truncation radius is 0.7 and that the halo concentration is 1, for the sake of simplicity. We compute 500 samples in each stacked profile and fit them with the TNFW and TNFWProb models. As shown in Table 6, the mean tangential profiles are able to recover the input values. In TNFWProb model fitting, since the mean and standard error of the truncation radii are degenerate, are not well constrained in some cases, resulting in a large mean measurement error.
Note. – aafootnotemark: : Number of simulated subhalos. bbfootnotemark: : Mean and standard error of the mass () and truncation radius () for simulated subhalos. ccfootnotemark: : Mean ratio of outputs to inputs for the TNFW model. Errors shown are the standard deviation based on 500 realizations. The values in brackets are the mean measurement uncertainties. ddfootnotemark: : Mean ratio of outputs to inputs for the TNFWProb model.
4. Cluster Galaxy - Galaxy Lensing Analysis
Galaxy-galaxy lensing analysis for member galaxies selected solely by their luminosities provides us with complementary and important information regarding cluster subhalos, because the sample is unbiased with respect to the lensing definition of subhalos (Sec 3). We use member galaxies with luminosities in the band larger than . We compute a stacked tangential distortion profile as a function of the radius from luminous member galaxies. Since the cluster field is crowded, neighboring luminous galaxies may lead to serious contamination in the stacked lensing profile, if they are not sufficiently separated. We thus need to determine the outermost radius of the profile in order to minimize lensing contamination from neighboring luminous members. We estimate the histogram of projected distances between the luminous member galaxies. The outermost radius is chosen to be by applying a threshold that the mean number of neighboring luminous galaxies is less than unity. We compile 64 luminous member galaxies located in the projected cluster-centric radius of . Figure 10 shows the mean tangential profile. A sharply truncated profile is not found, in contrast to lensing-selected subhalos (Figures 7 and 8). The NFW and TNFWProb models are then applied to describe the profile, and these two models (Table 7) give an acceptable fit. The best-fit tangential profile for the TNFWProb model is similar to that for the NFW model, because the intrinsic distribution of makes a sharply truncated profile blunt. The mean virial radius for the NFW model, , is larger than the mean truncation radius for the TNFWProb model. On average, three luminous galaxies are inside the mean virial radius of luminous galaxies. In other words, the mass distribution of subhalos associated with luminous galaxies overlap each other. The NFW model is therefore unlikely to represent halos associated with luminous member galaxies. On the other hand, the TNFWProb model gives a large scatter of the truncation radius compared to the mean value, . A broad distribution of the truncation radius smooths the truncation feature in the mean tangential profile for a large sample of less massive subhalos, which makes it difficult to resolve the subhalo size. To confirm this explanation, we conducted a stacked lensing analysis using a mock shear catalog (Table 6) in the same way as Section 3.4.5. We found that the truncation feature in the stacked profile is obscured and the mock simulation recovers the input values. Although we also conducted fitting stacked profiles for subsamples divided by luminosities or projected distances from the cluster center, only the upper limits can be derived. We also checked less luminous galaxies with luminosities less than , but found significant contamination from neighboring luminous or less luminous galaxies in the mean tangential profile.
Previous studies (Natarajan & Springel, 2004; Natarajan et al., 2007, 2009; Limousin et al., 2005, 2007) conducted galaxy-galaxy lensing studies for clusters at using single-band images. As described by (Broadhurst et al., 2005) and (Okabe et al., 2010b, 2013), lensing signals would be significantly diluted by a contamination of unlensed member galaxies in the shear catalog. Their catalog for background source galaxies using the single filter would suffer from a contamination of member galaxies. It is thus difficult to make a fair comparison between our result and the previous studies.
Note. – aafootnotemark: : Number of luminous member galaxies selected by luminosities () and cluster-centric radii (). bbfootnotemark: : Best-fit mass, the average and standard error of truncation radius distribution for the TNFWProb model. ccfootnotemark: : Best-fit virial mass and halo concentration for the NFW model. ddfootnotemark: : Average luminosity . eefootnotemark: : Signal-to-noise ratio for the tangential distortion profile.
5. Main Cluster Mass Measurement
A tangential distortion profile, , with respect to the cluster center, is a powerful tool to estimate the cluster mass (Okabe et al., 2010b). The tangential distortion profile as a function of the projected cluster-centric radius is computed by azimuthally averaging the measured galaxy ellipticities. Here, we assume that the cluster center is at the central position of the subhalo “21” associated with the cD galaxy (NGC4874). The top panel of Figure 11 shows a complex feature of the lensing profile in the radial range of , extending over 2 orders of magnitude. Here, the data points are calculated using the shear catalog without the LSS modeling. The lensing signal changes from to as the radius increases. As expected from the low lensing efficiency of the nearby cluster, the lensing signal is 1 order of magnitude lower than that for massive clusters at (e.g., Okabe et al., 2010b). However, the S/N reaches S/N thanks to a remarkably large number of background galaxies, which is comparable to or higher than those of clusters at (Okabe et al., 2010b). This high S/N validates weak-lensing analysis for low redshift clusters () which have been overlooked for a long time. We also find that the rotated component, (bottom panel of Figure 11), which is a non-lensing mode serving as a monitor of systematics errors, is 1 order of magnitude smaller than the lensing mode, , which is consistent with a null signal.
The tangential distortion contains complete information for the lensing signals, including the smoothed mass component of the main cluster, the interior substructure (Section 3.2) and LSS lensing signals behind the cluster (Section 2.6). To understand the profile, we computed the tangential shear profiles for 32 shear-selected subhalos and the LSS lensing model, illustrated by the blue dotted and magenta dashed-dotted lines in the top and middle panels of Figure 11. The S/N for the subhalos is S/N , accounting for of the total distortion signal. This indicates that the profile highly resolves the lensing signal from the interior substructure by the large apparent size of the cluster. The observed signal in the central region () is dominated by the subhalo “21”. The lensing signals in and in are depressed by prominent subhalos. As for LSS lensing, the signal-to-noise ratio is S/N. Here, since the LSS lens modeling for possible background groups, “F” and “I” (Table 2 and Figure 3), has failed significantly, we estimated the lensing distortion pattern from the NFW profile determined by the tangential profile for this object and found that this background group depresses the observed lensing signal at .
We fit a single NFW model to the distribution profile corrected using the LSS lensing model. The lensing signals from the subhalos gradually change from positive to negative in the central region (). These absolute values account for a large fraction of the total lensing signals, which makes it difficult to discriminate between signals from the subhalos and the main cluster based on the central signals. Indeed, when a single NFW model fits the data, the best-fit halo concentration is systematically changed () by a choice of the innermost radius. To avoid the subhalo bias in cluster mass measurements, we estimate the radial range for the fitting, by requiring that the fraction of the absolute value of subhalo signals to observed signals is less than . We fit the tangential profile between () and () with a single NFW model. Here, the physical scale of the innermost radius is comparable to those for massive clusters at (Okabe et al., 2010b). The resultant masses at different overdensities are listed in Table 8. The virial mass and concentration are and , respectively.
We also fit the profile in the full range with the NFW model for the smooth matter component and a central point mass contribution of the brightest cluster galaxy (BCG). We obtain the point mass , which is consistent with the projected mass measurement of subhalo “21” (Table 3). The summation of the virial mass and the point mass is . The total mass is in agreement with the estimated virial mass derived using the tangential fit for the radial range to minimize subhalo contributions.
We next repeat fitting with the NFW model as the smooth component, by fully taking into account lensing signals from all shear-selected subhalos and the LSS lensing model. The best-fit profile for the smooth component is shown in the green dashed line in Figure 11. The total signal (red solid line) from three different components of the smooth NFW profile, subhalos and LSS lens model describes the observed signals remarkably well. The summation of the virial mass and subhalo masses, , is in good agreement with the virial mass (Table 8). A singular isothermal sphere (SIS) model is strongly disfavored as the smoothed mass component, returning a goodness-of-fit statistic of .
We also conduct the cluster mass measurement using the LSS error covariance matrix (e.g., Schneider et al., 1998; Hoekstra, 2003), as described in Section 3.4.2. Although the error covariance matrix does not significantly change the result of subhalo mass measurement, the situation for the cluster mass measurement is slightly different. The diagonal component of the LSS error covariance matrix becomes comparable to the statistical error for . The S/N estimated from the covariance matrix in the tangential shear profile becomes smaller from the case of the statistical errors, because the LSS errors in different radial bins are correlated. The S/N is consistent with Hoekstra (2003). We use the shear catalog without the correction of the LSS model in the full radial range of the cluster and apply the NFW model for the smooth matter component and a central point source of the BCG. The best-fit mass is . The upper and lower errors become larger by and , respectively.
The projected mass () measurement ( statistics) for the main cluster is less sensitive to lensing signals from subhalos, because it estimates a cumulative profile. It is thus complimentary to the tangential fit. Figure 12 shows the profile calculated with the fixed background annulus of . The background region is inside the best-fit virial radius ( arcmin) derived from the tangential shear fit. Following Okabe & Umetsu (2008), we fit the profile with a single NFW model, taking into account the error covariance matrix. The best-fit profile is shown by the red solid line. The best-fit values (Table 8) are consistent with those for the tangential shear fit.
We compare the best-fit mass and concentration with results in the literature (Table 9). The mass measurements in this study are consistent with our previous analysis (Okabe et al., 2010a). The statistical precision of the new mass estimates is improved by four times thanks to the huge number of background galaxies. Gavazzi et al. (2009) conducted fitting the tangential distortion profile in the range of using the NFW model including and excluding priors on mass and concentration relations, using CFHT/Megacam data. In that study, the LSS lensing effect was not accounted for. The best-fits, regardless of priors, are compatible within their large errors, with the present study. However, their lensing signals (Figure 3 in Gavazzi et al., 2009) differ from the results in the present study (Figure 11). Their profile () is well described by a single NFW model. The lensing signals in the same radial range in the present study are dominated by prominent subhalos in and by the smooth mass component in , respectively. We conduct the tangential fit using the profile computed with the same radial bins as Gavazzi et al. (2009) and a single NFW model and obtain only an upper limit on because of an inadequate model. Kubo et al. (2007) carried out weak-lensing analysis using SDSS data. The best-fit (Kubo et al., 2007) derived by fitting the tangential profile up to with a single NFW model is three times higher than that observed here. The background LSS lensing effect was not accounted for in that study. Since their outermost radius () is five times higher than our best-fit virial radius, their mass would be overestimated by mass distribution outside the cluster. The dynamical mass estimates of and (Rines et al., 2003; Łokas & Mamon, 2003) agree with our best-fits, although their concentration is three times higher than estimates in the present study.