A&A 455, 441-451 (2006)
M. Kleinheinrich1,2 - P. Schneider2 - H.-W. Rix1 - T. Erben2 - C. Wolf3 - M. Schirmer2 - K. Meisenheimer1 - A. Borch1 - S. Dye4 - Z. Kovacs1 - L. Wisotzki5
1 - Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
2 - Institut für Astrophysik und Extraterrestrische Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
3 - Department of Physics, Denys Wilkinson Bldg., University of Oxford, Keble Road, Oxford, OX1 3RH, UK
4 - School of Physics and Astronomy, Cardiff University, 5 The Parade, Cardiff, CF24 3YB, UK
5 - Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
Received 23 December 2004 / Accepted 19 April 2006
We present mass estimates for dark matter halos around galaxies from the COMBO-17 survey using weak gravitational lensing. COMBO-17, with photometry in 17 optical filters, provides precise photometric redshifts and spectral classification for objects with R<24. This permits to select and sort lens and source galaxies by their redshifts and lens luminosity or color, which bypasses many uncertainties in other weak lensing analyses arising from broadly estimated source and lens redshifts. We study the shear created by dark matter halos around 12 000 galaxy lenses at redshifts by fitting the mass normalization of either singular isothermal spheres (SIS) or Navarro-Frenk-White (NFW) profiles to background source orientations around the whole lens sample. We also consider halos around blue and red subsamples separately and constrain the scaling of halo mass with light. For the NFW model, we find virial masses for blue and for red galaxies of , respectively. The 1- uncertainty on for the whole lens sample is about 0.2. We compare our results to those obtained from the Red-Sequence Cluster Survey (RCS) and the Sloan Digital Sky Survey (SDSS). Taking differences in the actual modelling into account, we find very good agreement with these surveys.
Key words: gravitational lensing - galaxies: fundamental parameters - galaxies: statistics - cosmology: dark matter
Several methods have been applied to measure masses of galaxies: rotation curves of spiral galaxies provided the first evidence for dark matter in galaxies (e.g. Sofue & Rubin 2001). In elliptical galaxies, the dynamics of e.g. the stellar population itself, globular clusters or planetary nebulae can be used (e.g. Danziger 1997). However, these methods can only be applied over radii where luminous tracers are available (a few tens of kpc) and suffer from typical problems of dynamical studies, e.g. the unknown degree of anisotropy (Rix et al. 1997).
In a few cases it is possible to measure masses from X-rays or, within the Einstein radius, from strong lensing (e.g. Kochanek et al. 2004).
Only two methods are currently in use to probe dark matter halos of galaxies at scales of about or larger: weak gravitational lensing and the dynamics of satellite galaxies. At these scales, the baryonic contribution to the mass is negligible, so that basically just the dark matter around galaxies is probed. In the beginning, satellite dynamics was only applied to isolated spiral galaxies (Zaritsky & White 1994; Zaritsky et al. 1997,1993). Later, early-type galaxies were investigated (Prada et al. 2003; McKay et al. 2002; Brainerd & Specian 2003; Conroy et al. 2005), and now galaxies in denser environments and at much fainter magnitudes are also studied (van den Bosch et al. 2004). Weak gravitational lensing has become a standard tool in recent years. Its main advantage over satellite dynamics it that no assumptions on the dynamical state of the galaxies under consideration have to be made.
Galaxy-galaxy lensing is the technique which uses the image distortions of background galaxies to study the mass distribution in foreground galaxies. Galaxy-galaxy lensing and satellite dynamics are independent methods and it is very desirable to have both methods available for comparison of results. Weak gravitational lensing is similar to the use of satellite dynamics in the sense that only statistical investigations are possible due to the weakness of the gravitational shear and the small number of satellites per primary galaxy. The most recent results from both techniques are summarized in Brainerd (2004).
Galaxy-galaxy lensing was measured for the first time by Brainerd et al. (1996). In early work, people concentrated on isothermal models for describing the lenses (Hoekstra et al. 2003; Brainerd et al. 1996; Griffiths et al. 1996; Wilson et al. 2001; Hudson et al. 1998; dell'Antonio & Tyson 1996; Fischer et al. 2000; Smith et al. 2001). The best-constrained parameter was the effective halo velocity dispersion for galaxies, assuming a scaling relation between the velocity dispersion and the luminosity according to the Tully-Fisher and Faber-Jackson relations. The Tully-Fisher index in this relation (see Sect. 4) was typically assumed according to measurements from the central parts of the galaxies. Only Hudson et al. (1998) were able to put at least a lower limit on . More generally, the galaxy-mass correlation function was later investigated (Sheldon et al. 2004; McKay et al. 2001) and the Navarro-Frenk-White (NFW) profile was considered (Hoekstra et al. 2004; Seljak 2002; Guzik & Seljak 2002). The Sloan Digital Sky Survey (SDSS) has turned out to be extremely powerful for galaxy-galaxy lensing studies (Seljak 2002; Sheldon et al. 2004; Seljak et al. 2005; McKay et al. 2001; Fischer et al. 2000; Guzik & Seljak 2002). Most of its success is due to the fact that the SDSS provides a large sample of lens galaxies with measured spectra. The spectra provide very accurate redshifts and classification of the lens galaxies. Therefore, it is possible to measure dark matter halos as function of luminosity, spectral type or environment of the galaxies. However, the SDSS is a very shallow survey and is thus only able to measure lens galaxies around . The question of halo properties at higher redshift and of evolution cannot be addressed. Therefore, substantial effort goes into measurements of galaxy-galaxy lensing at higher redshift with at least rough redshift estimates for the lenses from e.g. photometric redshifts. Currently available are the Hubble Deep Fields which are far too small to provide statistically clean samples free from cosmic variance. Wilson et al. (2001) addressed the question of evolution, but only for early-type galaxies. Hoekstra et al. (2003) had redshifts available for part of their lenses, but not yet enough to split the lenses into subsamples and to study their properties separately. Hoekstra et al. (2004) use the larger Red-Sequence Cluster Survey (RCS) with deep observations. No redshift or color information is available, yet, so that again only properties averaged over all classes of galaxies can be investigated.
Here, we will use the COMBO-17 survey which is a deep survey, that provides accurate photometric redshifts and spectral classification from observations in a total of 17 filters (Wolf et al. 2003,2004,2001). This data set allows us to probe lens galaxies at higher redshift ( ), to derive the relation between luminosity and velocity dispersion (or mass) instead of assuming it and to measure dark matter halos for blue and red galaxies separately. In addition to the singular isothermal sphere (SIS) we will also apply the NFW profile (Navarro et al. 1995,1996,1997).
This paper is organized as follows: in Sect. 2 we briefly describe the data set and in Sect. 3 our method of measuring galaxy-galaxy lensing. Section 4 gives our results from the SIS model and Sect. 5 those from the NFW profile. In Sect. 6 we investigate how our measurements are affected by the presence of large foreground clusters in one of the survey fields. In Sect. 7 we will compare our results to those from the RCS and the SDSS. We close with a summary in Sect. 8. Throughout the paper we assume and .
COMBO-17 covers three disjoint fields of each. Only one of the survey fields is a "random field''; a second is centered on the Chandra Deep Field South, and the third was chosen to study the supercluster system Abell 901a/b, Abell 902 and represents thus an overdense line-of-sight. In Sect. 6 we will investigate the influence of the supercluster system on the galaxy-galaxy lensing measurement. The average of these three fields should not be too far from cosmic average.
The data set used here is exactly the same as in the method-focused companion paper Kleinheinrich et al. (2005). We refer the reader to this paper for details on the COMBO-17 survey and the shape measurements and just summarize the most relevant points. The COMBO-17 (Wolf et al. 2003, 2004) project provided deep R-band imaging ( ) at a seeing of ( ) over three disjoint fields of about 0.26 square degrees each, along with redshift estimates from 17 band photometry, which are accurate to (at R<21) and to at the redshift sample limit of . The R-band images, which are the basis of the image shape measurements for the lensing analysis, were augmented with public archival ESO data, and reduced with a pipeline that was optimized for weak lensing (Schirmer et al. 2003). Shapes, expressed by the complex ellipticity , were measured with a variant of the KSB algorithm (Luppino & Kaiser 1997; Kaiser et al. 1995; Hoekstra et al. 2000), as implemented by Erben et al. (2001).
|Figure 1: Left panel: averaged tangential shear as a function of projected separation between lens and source; right panel: same as left panel but sources being rotated by 45 degrees.|
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We bin the lens-source pairs by their projected separation r at the redshift of the lens and calculate a weighted mean tangential shear for all pairs per bin. The weighting takes into account only the noise in the psf-corrected ellipticities of the sources. The weight of each source is taken as the inverse of the variance of the N=20 objects being closest to the source in - S/N space. Here, is the half-light radius and S/N is the signal-to-noise of the source - see Erben et al. (2001) and Kleinheinrich et al. (2005) for details on this weighting scheme. In particular, no weighting according to the luminosity of the lens is applied. The error bars are obtained from repeating this measurement a hundred times with randomized source orientations.
The right panel of Fig. 1 was obtained in the same way as the left panel, but after rotating the sources by 45 degrees. The signal is consistent with zero indicating that the measured tangential shear is indeed due to gravitational lensing.
Note that we do not use the measurement of the mean tangential shear in our actual analysis of the data, as this approach does not exploit e.g. the redshift and luminosity information we have for lenses. However, the plots shown in Fig. 1 might be helpful for comparison with results from other data sets.
Because the shear from individual galaxies is weak and because
several halos may contribute comparably to the shear of a given
source position we sum the shear contribution from different
lenses to derive the total shear acting on source j:
In this approach the most likely model is one that produces shear-corrected source orientations that are isotropically oriented.
Note that in this modelling approach the common statistic of shear strength vs. projected radius (shown in Fig. 1) is never used, as we use a S/N-optimized individual shear estimate for each image, which depends on the global lens model.
With the COMBO-17 data set we can select lenses and sources based on their redshifts. We only use objects classified as galaxies or likely galaxies by COMBO-17. Both lenses and sources lie in the magnitude range R=18-24 (Vega). The bright limit was set to avoid saturation, while the faint limit is given by the magnitude limit down to which the classification and redshift estimation in COMBO-17 works reliably. Note that with such redshifts we are able to include faint lenses and bright sources.
Our lens sample consists of all galaxies with redshift . Lenses with higher redshifts would not add much to the constraints because only a small number of sources would lie behind them. Lenses at smaller redshifts are excluded because for them a given corresponds to a large angular separation . Because we exclude all sources for which lenses might lie outside the field boundaries we would therefore have to exclude too many sources.
|Figure 2: Constraints on the velocity dispersion and the Tully-Fisher index for the SIS model with using all lenses ( left panel) and subsamples of blue and red lenses ( right panel). Contours are 1-, 2- and 3-.|
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We chose sources with individual redshift estimates in the redshift range , although in principle we could also include sources for which only statistical redshifts are available. However, in Kleinheinrich et al. (2005) we found that the inclusion of these sources does not improve the constraints, probably because sources that are too faint to get individual redshift estimates also have too noisy shape measurements. A potential lens-source pair is only evaluated if . The minimum redshift difference of dz=0.1 is chosen to remove pairs where, due to redshift errors, the source is actually in front of the lens or physically close to it. Thus, we minimize the contamination from intrinsic alignments (King & Schneider 2002; Heymans & Heavens 2003).
In the absence of survey image edges and systematic shape measurement errors, there is no a priori need to set a value for ; the large number of images with tiny shear contributions from a given lens galaxy should still add signal to the estimate. However, in practice a choice of is sensible for a variety of reasons. First, a SIS model has a diverging cumulative mass profile that is unphysical. Second, a very large value of excludes too many source images from the analysis due to edge effects. Third, at separations of our underlying halo model assumption, that halos are spherical, isolated and spatially uncorrelated, breaks down. Therefore, we probe lens galaxy halos out to some projected, physical distance from their center. With individual lens redshifts, this can be converted to an angular separation that can vary among different lens-source pairs. When working with the SIS model (Sect. 4) we use , and . Note that for radii larger than , there is evidence (e.g. Prada et al. 2003) from satellite dynamics, that a SIS is a poor approximation to the dark matter profile. When modelling the lenses by NFW profiles we use a maximum separation of to ensure that the region around the virial radius is probed even for the most massive galaxies.
Shapes can only be measured reliably if objects do not have nearby images. Therefore, we only use lens-source pairs with a minimum angular separation . At this translates into a physical separation , at into .
In addition to fitting halo properties by averaging over all lenses, we will also consider blue and red subsamples based on rest-frame colors. The definitions of "red'' and "blue'' are based on the "red sequence'' of galaxies in the color-luminosity plane, found to be present to in COMBO-17 (Bell et al. 2004). Galaxies with define the blue sample while all other galaxies are in the red sample (Bell et al. 2004). In total, our data set contains 11 230 blue and 2580 red lens candidates. The average redshift and SDSS r-band rest-frame luminosities are and for the blue sample and and for the red sample, respectively.
The (aperture) mass-to-light ratio of the galaxies is determined
For each lens-source pair the shear amplitude is given by
Table 1: Results for the SIS model for different lens samples and . , and are the numbers of lenses, sources and pairs in each measurement. Note that the number of sources can vary for different lens selections because not all source candidates are always lying within of a lens. The best fit values of and are given with 1- error bars. The last two columns give the aperture mass within and the corresponding mass-to-light ratio.
We also explored values of larger than . For we find and while for we find and . There is a clear decrease in when larger scales of the galaxy halos are probed. Further, we see a systematic increase of with scale. These trends of and with scale show that the model adopted here to describe the lenses is too simple. Several causes are possible: the density profiles of dark matter halos might decline more steeply than r-2 as also implied by satellite dynamics (Prada et al. 2003). Alternatively, the environment of the lens galaxies might contribute differently on different scales or the contribution of different subclasses of lenses might change with scale. This all clearly shows that when comparing results from different galaxy-galaxy lensing studies it is important to compare measurements obtained at similar scales.
The shear from the NFW model is calculated in Bartelmann (1996) and Wright & Brainerd (2000). Formally, NFW profiles are a two-parameter family, e.g. with and c as possible choices. Yet, for a given cosmological model (e.g. CDM), redshift (e.g. ) and halo mass scale, N-body simulations show that c has only little scatter.
|Figure 3: Constraints on the virial radius and the concentration c for the NFW profile obtained from the full lens sample with fixed and . The viral radius and the concentration c are defined as detailed in Sect. 5.1.|
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First, we try to constrain the virial radius and the concentration c for a fixed . Figure 3 shows results for the whole lens sample with and . Using instead does not change the shape of the contours but just shifts them toward lower values of . As Fig. 3 shows, weak lensing provides only poor constraints on c: only a lower bound can be obtained. The 1- lower limit for c alone is c>29, the 2- lower limit is c>11. In the following, we will use a fixed . This choice is somewhat arbitrary. However, smaller values of c are disfavoured by our data. We checked that assuming a smaller c does not change the best-fit results of our fits but would increase the error bars.
For the NFW model with we obtain a scaling of the virial mass-to-light ratio with luminosity as implying almost the same mass-to-light ratio for all luminosities. In contrast, we find a decreasing mass-to-light ratio at fixed radius with increasing luminosity for the SIS model. However, the scaling relation found from the SIS model is only marginally excluded at the 1- level by the measurement from the NFW model.
|Figure 4: Constraints on the virial radius and its scaling with luminosity () inside for the NFW profile using all lenses ( left panel) and subsamples of blue and red lenses ( right panel). Contours are 1-, 2- and 3-.|
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Table 2: Constraints on dark matter halos of galaxies modelled by NFW profiles for different lens samples and . , and are the numbers of lenses, sources and pairs in each measurement. These numbers differ from those given in Table 1 because of the larger used here. The virial radius and are fitted quantities (see Fig. 4), the virial mass and the virial mass-to-light ratio are calculated from . gives the scaling of with luminosity, . All errors are 1-.
The S 11 field contains the cluster Abell 1364 at a redshift z=0.11 for which we fit a velocity dispersion . As for the foreground clusters in the A 901 field, we compare measurements for the S 11 field, first including and then ignoring the cluster shear. We find that the cluster shear is negligible for the S 11 field.
Table 3: Central positions, redshifts and velocity dispersions of the known clusters in the A 901 field (Taylor et al. 2004).
Because only lens galaxies with have been considered, the lens sample should contain no galaxies that lie in the foreground clusters (Abell 901a/b, Abell 902). The expectation is that the foreground clusters will not influence the galaxy-galaxy lensing measurements because the shear from the foreground clusters will be in random directions with respect to the orientations of the lens-source pairs used for galaxy-galaxy lensing.
The left panel of Fig. 5 shows likelihood contours obtained from the A 901 field both ignoring and including the shear from the foreground clusters Abell 901a/b and Abell 902. The shear from CBI is ignored. Hardly any difference is seen between both cases, from which we measure and (1- errors).
|Figure 5: Influence of the clusters shear on the galaxy-galaxy lensing measurement. Solid lines refer to measurements ignoring the cluster shear, dashed lines to measurements including the cluster shear. In the left panel, only the foreground clusters Abell 901a/b and Abell 902 are taken into account. In the right panel also the background cluster CBI is used. A maximum projected separation between lenses and sources of is used.|
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As a second test we ignore the cluster shear but exclude all sources from the galaxy-galaxy lensing measurement for which the convergence (and thus the shear ) from the foreground clusters exceeds some threshold. Figure 6 shows that from the foreground clusters alone for almost all sources. The maximum is . Therefore the weak shear limit is still valid in Eq. (14). When excluding all sources with (224 out of 6481 sources), the difference to the measurement using all sources is negligible. When excluding further sources, the contours start to widen significantly but maintain the same minimum. This widening can be explained by the decreasing number statistics. In particular, we do not see any hint of a bias from the clusters. When excluding the 22% of sources with , the 3- contour remains closed. The best-fit parameters with 1- errors are then and .
|Figure 6: Histogram over due to the foreground clusters only (solid) and due to the foreground clusters and CBI together (dashed).|
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The right panel of Fig. 5 shows the same as the left panel but this time includes the shear from CBI in the cluster measurement. Although the change in the contours is clearly visible, the shear from the background cluster CBI does not seem to have a big influence; the contours only widen marginally. The best-fit parameters with their 1- errors become and . Note, however, that the resolution of our grid in parameter space is only .
As a second test we again ignore the cluster shear and exclude objects with large from both the foreground clusters and CBI. Figure 6 shows a histogram of , the maximum of which is . When excluding all sources with we obtain likelihood contours similar to the case when including the cluster shear. 345 out of 6481 sources are then excluded. Excluding further sources results in wider contours. The 3- contour stays closed when we exclude sources with . These are 22% of all sources and the contours do not shift compared to the case where the cluster shear is ignored.
A third test is to exclude all lenses that are close to the center of the background cluster CBI. We exclude every lens lying within projected separation from the cluster center (corresponding to angular separation) and with a redshift difference less than 0.2. These are 221 out of 4444 lenses. The likelihood contours hardly change when these lenses are excluded.
From all these tests we conclude that our results are not biased from the presence of several clusters in the A 901 field. This conclusion also holds when the SIS model is fitted within instead of . The little influence even the background cluster CBI has might be surprising but it is understandable given the small number of lenses that actually reside within this cluster. In particular we note that the comparably large value of obtained from the A 901 field which is about 1- higher than from all three fields together is not caused directly by the clusters. Instead, it seems that cosmic variance is still an issue even on the scales of our survey fields.
However, crucial differences exist. The area of the RCS used for the galaxy-galaxy lensing analysis is , so about 60 times larger than that used here. On the other hand, Hoekstra et al. (2004) only have observations in a single filter available. Therefore, they have to select lenses and sources based on magnitude cuts and they have to use redshift probability distributions for estimating luminosities and for shear calibration. Furthermore, Hoekstra et al. (2004) can only investigate lens galaxy halos on an angular scale while we are able to probe the same physical scale of all lenses. Third, there is no easy way to implement halo scalings with luminosity, nor to separate lenses by their rest-frame color. Hoekstra et al. (2004) fit their models within 2' which they estimate corresponds to about at the mean redshift of the lenses. This is considerably larger than the region we probe for the SIS model ( ) and comparable to the region probed for the NFW profile ( ). Another difference lies in the definition of the fiducial luminosity. We use measured in the SDSS r-band while Hoekstra et al. (2004) use as reference. From our data set we calculate that lenses with have SDSS-r-band luminosities of about . In Kleinheinrich et al. (2005) it is shown for the SIS that it is not possible to constrain the scaling relation between velocity dispersion and luminosity without multi-color data. Therefore, Hoekstra et al. (2004) have to assume this scaling relation unlike being able to fit it as we do.
Hoekstra et al. (2004) do not specifically fit the SIS model using the maximum-likelihood technique of Schneider & Rix (1997). Instead, they fit the SIS model to the measured galaxy-mass cross-correlation, and they constrain the truncated SIS model using the maximum-likelihood technique. The velocity dispersion obtained from the truncated SIS model becomes that of the SIS in the limit of infinite truncation parameter s. In both cases, Hoekstra et al. (2004) assume a scaling relation as in Eq. (5) with and a characteristic luminosity as detailed in Sect. 7.1. From the galaxy-mass cross-correlation function they obtain . The truncated SIS yields . Both results are in good agreement with our results. Conversely, we are able to confirm the value of adopted by Hoekstra et al. (2004).
The error bars of Hoekstra et al. (2004) are about 5 times smaller than ours. From the difference in area between the RCS and COMBO-17 one would even expect a factor of about difference. That their error bars are not that much smaller can be attributed to the detailed classification and redshifts available in COMBO-17. In Kleinheinrich et al. (2005) it is shown that the error bar on increases by about 30% when redshifts are omitted. The influence of accurate redshift estimates on the determination of is found to be much more severe - from just a single passband no meaningful constraints on can be derived. This explains why we are able to fit while Hoekstra et al. (2004) had to assume a fixed value.
The exact modelling of Hoekstra et al. (2004) is somewhat different from ours. While we fit the virial radius and its scaling with luminosity, Hoekstra et al. (2004) assume scaling relations between the maximum rotation velocity and virial mass, respectively, with luminosity and fit the virial velocity and the scale radius rs. The virial velocity is directly related to the virial mass and virial radius by . All three quantities are independent of the scale radius rs. Therefore, we can readily compare our results on the virial mass to those from Hoekstra et al. (2004). They find in good agreement with our result. Furthermore, their constraint on the scale radius is roughly consistent with the scale radius implied by the virial radius we find and the concentration we assume.
Guzik & Seljak (2002) use the halo model for their galaxy-galaxy lensing analysis. This model takes not only the contribution from galaxies themselves into account but also from the surrounding group and cluster halos. This of course complicates the comparison. On the other hand, Guzik & Seljak (2002) find that on average only about 20% of the galaxies are non-central galaxies, for which the group/cluster contribution is important. Therefore, the group/cluster contribution to our mass estimates is probably well within our error bars. Guzik & Seljak (2002) only fit NFW profiles to their data. They define the virial radius as radius inside which the mean density is 200 times the critical density of the universe. Therefore, we have to multiply our virial masses by a factor 0.79 to compare them to theirs. Similar to our approach, Guzik & Seljak (2002) assume a relation between mass and luminosity . and are derived from different passbands. Here, we just compare to the results from the r'-band in which the reference luminosity is .
Using a model to link lens galaxies to their dark matter halos, either SIS or NFW, we constrained the halo parameters through galaxy-galaxy lensing. For both halo models, we find that, at the same luminosity , red galaxies have dark matter halos about twice as massive as blue ones. For SIS models within we obtain for blue galaxies and for red galaxies. Adopting NFW profiles we find virial masses for blue galaxies and for red galaxies. Note that the virial masses are defined as masses inside a sphere with mean density equal to 200 times the mean density of the universe. For this is about . Changing this definition to the mass inside a sphere with mean density equal to 200 times the critical density of the universe would lower the virial masses by about 20%. The mass estimates from the SIS model are considerably larger than from the NFW profile although the virial radii exceed the aperture adopted for the mass estimate of the SISs. However, it has been shown by Wright & Brainerd (2000) that such behaviour is expected.
For both models, we also fit the scaling between mass and luminosity. In the SIS model we find approximately the same scaling for blue and red galaxies, about inside . For the NFW model we find a similar scaling relation for blue galaxies, but for red galaxies. The differences between the two models and between the two subsamples for the NFW model may arise from the different scales over which the fitted mass-luminosity relation applies. For the SIS model it is always fitted over a fixed aperture of while for the NFW profile it only applies to the mass within the virial radius that differs between blue and red galaxies.
Finally, we compared our results to those obtained from the RCS and the SDSS. We pointed out that for such a comparison it is indispensable to compare results from similar modellings. As far as possible we translated our measurements to the modelling approaches of Hoekstra et al. (2004) for the RCS and Guzik & Seljak (2002) for the SDSS and found consistent results. We deem this remarkable, at least because the modelling of Guzik & Seljak (2002) is very different from ours. A more exact comparison between COMBO-17 and the SDSS adopting the same techniques is beyond the scope of this paper, although such an investigation would be very valuable. The fact that both surveys probe galaxies in different redshift ranges would in principle allow a measurement of the evolution of dark matter halos.
We thank the anonymous referee for helpful suggestions. C.W. was supported by a PPARC Advanced Fellowship. M.K. acknowledges support by the BMBF/DLR (project 50 OR 0106)), by the DFG under the project SCHN 342/3-1, and by the DFG-SFB 439.