A&A 455, 441-451 (2006)
DOI: 10.1051/0004-6361:20042606
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
Abstract
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).
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Figure 1:
Left panel: averaged tangential shear
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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.
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Figure 2:
Constraints on the velocity dispersion
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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
by :
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.
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Figure 3:
Constraints on the virial radius
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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.
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Figure 4:
Constraints on the virial radius
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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).
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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
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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
.
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Figure 6:
Histogram over ![]() |
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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.
Acknowledgements
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.