D. Clowe1,2, - G. A. Luppino2 - N. Kaiser2
1 - Institut für Astrophysik und Extraterrestrische Forschung der Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
2 - Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
Received 2 May 2003 / Accepted 21 July 2003
We use weak lensing shear measurements of six z>0.5 clusters of galaxies to derive the mean lensing redshift of the background galaxies used to measure the shear. Five of these clusters are compared to X-ray mass models and verify a mean lensing redshift for a 23<R<26.3, R-I<0.9 background galaxy population in good agreement with photometric redshift surveys of the HDF-S. The lensing strength of the six clusters is also analyzed as a function of the magnitude of the background galaxies, and an increase in shear with increasing magnitude is detected at moderate significance. The change in the strength of the shear is presumed to be caused by an increase in the mean redshift of the background galaxies with increasing magnitude, and the degree of change detected is also in agreement with those in photometric redshift surveys of the HDF-S.
Key words: cosmology: observations - dark matter - gravitational lensing - galaxies: distances and redshifts - galaxies: clusters: general
Weak gravitational lensing, where one measures the mass of a foreground object by detecting deviations from an isotropic background galaxy ellipticity distribution, can be used to obtain an independent estimate of the mean redshift of a galaxy population. Because the strength of the lensing signal varies with both the redshift of the background galaxies and the redshift of the lensing object, comparing the lensing strength of different populations of objects both within a given field and across different fields lensed by varying redshift foreground objects can be used to determine the mean redshift of the galaxy populations. This was attempted by Smail et al. (1994) using a set of three clusters at z=0.26, 0.55, and 0.89. Based primarily on the lack of lensing observed in the high redshift cluster, the data resulted in a best fit for a no evolution model where the majority of the I<25 galaxies were at z<1. It was later determined, however, that the z=0.89 cluster used had a very low X-ray luminosity (Castander et al. 1994). If the low X-ray luminosity is interpreted as a low mass, the lack of a weak lensing signal by this cluster would no longer constrain the faint galaxies to be at low redshift.
A weak lensing signal was detected in the high-redshift cluster MS 1054-0321, at z=0.826, by Luppino & Kaiser (1997), which implied that a large fraction of the galaxies must be at z>1. With the goals of determining the mass and dynamical state of X-ray selected, high-redshift clusters of galaxies and determining the mean redshift of the faint blue galaxy (FBG) population, we have undertaken a survey of six z>0.5 clusters. We selected as our sample of clusters the five EMSS high-redshift clusters (MS 0015.9+1609 at z=0.546, MS 0451.6-0305 at z=0.550, MS 1054.4-0321 at z=0.826, MS 1137.5+6625 at z=0.782, and MS 2053.7-0449 at z=0.583), which were the only z>0.5 clusters published from a serendipitous X-ray survey at the time, and one from the ROSAT North Ecliptic Pole survey (RXJ 1716.6+6708 at z=0.809) which was discovered shortly after we began our survey (Henry et al. 1997; Gioia et al. 1999). The weak lensing analysis of the clusters have been published (Clowe et al. 1998,2000). In this paper we present the results of our attempts to measure the mean redshift of the FBG population from their weak lensing signal.
In Sect. 2 we present the weak lensing techniques used in our analysis. Comparison of the weak lensing signal and X-ray mass estimates is given in Sect. 3. In Sect. 4 we present the results of direct comparison of the lensing signal of various galaxy populations. Section 5 contains our conclusions. Throughout this paper, unless otherwise stated, we assume an universe, parameterize our results in terms of H0 = 100 h km s-1Mpc-1 and give all errors as 1.
For a single thin lens, such as a cluster of galaxies, the strength of the
lens is expressed in the dimensionless mass surface density ,
|Figure 1: Plotted above are the values of as a function of background galaxy redshift for four lens redshifts. The redshifts of the lenses (0.1, 0.25, 0.5, and 0.8) can be determined from where becomes 0. While the lower redshift lenses have only slowing varying over the expected faint galaxy redshift distribution (0.8-2), the higher redshift lenses still have being a strong function of background galaxy redshift.|
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Table 1: Summary of cluster data.
What is measured from the background galaxies, however, is not ,
the reduced shear g, which is related to the gravitational shear by
A similar equation can be calculated for the non-weak lensing case, when , however in this case the effective mean lensing redshift will be a function of the local mass surface density. Further, due to the competing effects of deflection and magnification of the background galaxies, the redshift distribution of a magnitude limited sample of the background galaxies will change with increasing (e.g. Dye et al. 2001). As a result, accurately comparing the mean lensing redshifts of two galaxy populations near the cores of massive clusters is much harder than at large distances from the cores, and is near impossible without some pre-existing knowledge of the mass distribution of the clusters.
As can be seen in Eq. (4), in the weak lensing limit a sheet of constant
density across the field can be added to the cluster surface density without
affecting the measured shear. As a result, one cannot determine
at a given point uniquely, but can only determine them to
within an unknown additive constant. As a result, a useful statistic to
use as a mass estimate is aperture densitometry (Fahlman et al. 1994; Clowe et al. 2000),
Further, in the non-weak lensing limit, in aperture densitometry is replaced by , and the resulting statistic is no longer measuring . The statistic is also no longer linear with , but can still be used to find a best fit by converting a mass profile, which must cover the same range in r as , to a reduced shear profile and calculating the resulting statistic to compare with the measured value. If, however, one does have a mass profile determined from an independent data set, one will typically get a higher signal-to-noise measurement by fitting the observed reduced shear profile directly with the mass profile converted to reduced shear profile via the fit parameter. In both cases, the fitting for the mean lensing redshift can only be done in regions with a sufficiently low and that the magnification and displacement of the background galaxies do not significantly alter the background galaxy redshift distribution.
|Figure 2: Shown above are the best fit values for the mean lensing redshift of the background galaxy population as a function of , which is the mean surface mass density in the negative annular region for (Eq. (7)), for the five clusters with complete X-ray data. The solid lines are the best fit, and the dashed, dot-dashed, and dotted lines are the 1, 2, and 3- deviations from the best fit respectively. The vertical lines, with associated error bars, are the surface density for the annular region calculated from the best-fit X-ray -model. The horizontal lines, with associated error bars, are the mean lensing redshift calculated from the HDF-S photometric redshift catalog of Fontana et al. (1999) using the same magnitude and color cuts as in the observed data.|
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The standard model used to fit the X-ray data of the clusters is the
-model, for which the mass enclosed in a sphere of radius r is
In the above comparison, all of the background galaxies were de-magnified
before applying magnitude cuts to the catalog
by the amount
by assuming that
the best fit
model to the measured reduced shear over
range to calculate
This is a first order correction to the magnification of the
observed background galaxy population, and thus should make the observed
population be on average the same as those observed in blank fields.
However, because the lensing strength, and thus the magnification, is
a function of the redshift of the galaxies, the average magnification which
we corrected for will be slightly too low for high redshift galaxies and
too high for low redshift galaxies. Thus, higher redshift galaxies will
still be slightly magnified and lower redshift galaxies will be slightly
de-magnified. This will result in
a slight overestimation of
by this method, but
from simulations we have determined this systematic error is an order of
magnitude below the random errors in the measurement. In future, larger
data-sets, this error could be minimized by binning the data by colors into
groups with similar redshifts.
|Figure 3: Shown above are the best fit values for from comparing the X-ray -models with the weak lensing shear profiles as a function of cluster redshift. The horizontal bars intersecting the error bars indicate how much of the error bar is due to the errors in the weak lensing mass measurement, with the remainder due to the uncertainties in the X-ray mass measurement. The four solid curves are the values of for background galaxies at redshifts of 1, 1.5, 2, and 3. The dashed curve, with error bars, shows the value of from the HDF-S photo-z galaxy catalog.|
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As can be seen in Fig. 2, if the -model of the X-ray clusters is extrapolated to determine the mean in the annular region subtracted in , the allowable value for is in good agreement with that calculated from photometric redshifts of galaxies with the same magnitude and color range in the HDF-S (Fontana et al. 1999). If one is going to extrapolate the X-ray model over the region containing the measured reduced shear, however, one will obtain both a better signal-to-noise and avoid the systematic error of assuming g is by fitting the reduced shear profile with the -model surface mass profile. In Fig. 3 are the best fit values of when fitting the shear and mass models over a range. The 300 h-1 kpc inner radius was chosen to avoid the large changes to the background galaxy redshift distribution which occurs due to the larger magnifications and displacements of the background galaxies near the cluster core.
For these fits, the -model was converted from to by the fit value , and used to calculate the reduced shear profile . The model's and profiles were then used to calculate and correct for the average magnification for each background galaxy as a function of distance from the cluster center. The magnitude corrected catalog then had the magnitude and color cuts applied to select the catalog used to measure the reduced shear. The measured reduced shear was then compared with the model using a statistic, which was minimized to find the best fit . The resulting measurements can then be converted to a for each cluster. For a broad background galaxy redshift distribution, the resulting is a function of the lensing cluster redshift due to the change in the with cluster redshift. The results are in good agreement with the photometric redshift distribution of faint galaxies from the HDF-S.
It should be noted that the mean lensing background galaxy redshift is a function of magnitude, color, size, and surface brightness cuts placed on the background galaxy catalog. Because the images for the five clusters used in this comparison are similar in exposure times and seeing, the weak lensing results all use the same background galaxy redshift distribution. In general, however, this will not be the case and the mean lensing redshifts as a function of cluster redshift shown in Fig. 3 will not be the mean lensing redshifts of the observations. For each observation, the mean lensing redshift would need to be computed from a redshift catalog by applying the same cuts as are used to select the background galaxies.
As was discussed in Sect. 2, for a high-redshift lens, the strength of the
shear acting on a background galaxy greatly depends on the angular distance of the
background galaxy. As the ellipticity induced in the galaxy by the
weak lensing shear is smaller than the typical intrinsic ellipticity of the
galaxy, one cannot use this to determine angular distances of individual galaxies.
One can, however, use this to measure the relative distances of two galaxy
samples provided each sample has enough galaxies to reduce the mean
intrinsic ellipticity of the sample well below the expected shear level.
Ideally one would choose the samples in some manner, such as by using
photometric redshifts, which would allow the galaxies inside each sample
to be at a similar distance. It is, however, still
possible to measure a mean angular distance ratio for two sets of galaxies,
each of which has a broad redshift distribution.
|Figure 4: Shown above are values for the mean shear for the background galaxies, divided into four magnitude bins (23-24, 24-25, 25-25.7, and 25.7-26.3), relative to the mean shear of the brightest magnitude bin. Only galaxies located further than 350 h-1 kpc from the cluster centers were used to compute the mean shear. The mean shear of the three clusters are given by the open circles and the mean shear of the three clusters are given by the open squares. The filled circles and squares are the expected shear levels based on the photometric redshifts of the HDF-S galaxies in the same magnitude bins for clusters at z=0.55 and z=0.8 respectively.|
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In the weak lensing limit, where , the shear acting upon a galaxy is a function of the lens mass, the galaxy position, the lens and galaxy redshifts, and the cosmological model. If the galaxy samples being compared have the same spatial distribution about a common lens, then the ratio of the mean shears is a function only of the redshift of the lens, the redshift distributions of the samples, and the cosmological model. If the magnification of the background galaxies is corrected for, the galaxy samples around different lenses of similar redshift can be coadded to improve the signal-to-noise of the mean shear ratio.
In Fig. 4 we show the relative strength of the mean shear signal for the three clusters and the three clusters in four magnitude bins. For both sets of clusters, the strength of the shear signal increases with increasing magnitude, with significances, calculated from Student's t-distribution, of 96.4%, 72.4%, and 96.0% for the clusters, clusters, and both sets combined respectively. This is consistent with the mean redshift of the background galaxies increasing with magnitude. Also shown in Fig. 4 are the shears which would be measured from the Fontana HDF-S photometric redshifts when using the same magnitude bins.
Due to increasing more rapidly for higher redshift lenses, one should, in theory, be able to use multiple lenses at different redshifts to obtain estimates for the redshift distribution of the background galaxies. This can be seen in Fig. 4 in which the difference in the lensing strength predicted by the HDF-S photometric redshift catalogs for the and lenses continues to increase with increasing magnitude of the background galaxies. This difference, however, is too small to measure with this data set. We estimate that we would need a data set ten times as large (60 clusters) with the same quality of data in order to successfully apply any of the techniques (e.g. Bartelmann & Narayan 1995) to measure the background galaxy redshift distribution.
One source of systematic error in the weak lensing mass estimates can be the dilution of the shear signal from blue cluster dwarf galaxies. The background galaxy catalogs were selected from all detected galaxies with 23<R<26.3 and R-I<0.9. The color selection removed the red-sequence of cluster ellipticals from the galaxy catalogs, but would have left some fraction of the bluer cluster galaxies. Cluster galaxies at are redder in R-I than those at . As a result, applying the same color cut to both sets of clusters would remove a greater fraction of cluster spirals from the background galaxy catalogs than from the catalogs. From number counts of dwarf galaxies in nearby clusters (e.g. Trentham 1998), we estimate that the weak lensing shear signal, and thus the derived masses, could be under-predicted by 10-20% for the clusters. This estimate, however, depends greatly on a lack of evolution in the number counts of dwarf galaxies compared to the cluster L* population.
We also compared the ratio of the shear signals as a function of magnitude, and demonstrate that the measured shear does tend to increase with increasing magnitude. The amount of the increase is again in good agreement with the photometric redshifts of the HDF-S. This result is also in agreement withthat of Hoekstra et al. (2000), who compared the relative lensing strength of galaxies in an HST mosaic of MS 1054.4-0321. The level of noise in our comparison, however, is too great to attempt to obtain a meaningful mean lensing redshift as a function of magnitude for the background galaxies.
We thank Gillian Wilson, Lev Koffman, Len Cowie, Dave Sanders, John Learned, and Peter Schneider for their help and advice. We also wish to thank Megan Donahue, Isabella Gioia, and J. Patrick Henry for sharing their X-ray data with us. This work was supported by NSF Grants AST-9529274 and AST-9500515, Nasa Grant NAG5-2594, ASI-CNR, and the Deutsche Forschungsgemeinschaft under the project SCHN 342/3-1.