4 Gravitational lensing analysis in MS 1008-1224

4.1 Mass reconstruction from weak shear

The weak distortion of background sources produced by gravitational lenses can be used to construct the projected mass distribution of the lens (see Tyson et al. 1990; Mellier 1999; Bartelmann & Schneider 2000). The excellent quality of the FORS1 data-set, especially the depth and the seeing, enabled us to accurately correct for many of the non-gravitational distortions of the image (PSF shear/smear) and also explore in some detail issues like fidelity of reconstructed mass features. To do this two teams, using different source selection criteria and different mass reconstruction schemes, independently produced maps of the mass distribution in MS 1008-1224.

4.1.1 2-dim mass reconstruction algorithms

Method 1

The IMCAT weak-lensing analysis package has been made publicly available at the URL http://www.ifa.hawaii.edu/~kaiser by Kaiser and his collaborators (Kaiser et al. 1995; Luppino & Kaiser 1997). The specific version used was the one modified and kindly made available to us by Hoekstra (see Hoekstra et al. 1998). A description of the analysis including measurement of the galaxy polarization, correction for smearing by and anisotropy of the PSF and the shear polarizability of galaxies and the expression for the final shear estimate have already been given by Hoekstra et al. (1998) and references therein and will not be repeated here.

The PSF anisotropy varied across the image and was about 0.015 (shape polarisation). The variation across the image was determined and corrected (separately in each band) using 50-60 stars scattered all over the CCD. The correction resulted in a mean residual polarisation of 0.0002 (rms = 0.004) for these PSF stars.

The maximum probability algorithm of Squires & Kaiser (1996), with K = 20(number of wave modes) and $\alpha = 0.05$ (the regularisation parameter), was used to reconstruct the mass distribution from the shear field. Our analysis differs from that of Hoekstra et al. only in the weighting of the data at the final stage (described next).

Error weighting:

The contribution of gravitational shear to the observed ellipticity of a background source is only a small fraction of its intrinsic ellipticity and so one has to average the shapes of many (10-20) sources in the neighbourhood to estimate the gravitational shear at that location. Further, the individual values have to be appropriately weighted to obtain meaningful results. Following Hoekstra et al., we estimated the average value of the shear at a location ${\left<\gamma\right>} = \Sigma(W_i\gamma_i$)/ $\Sigma W_i$, where $\gamma_i$ is the shear of individual galaxies and W_i = G_i/ $(\delta\gamma_i)^2$ is the weight comprising the error on the individual shear ( $\delta\gamma_i$) and a Gaussian factor G_i depending on the distance of the i-th galaxy from the location at which the average shear was being calculated. The typical value of $\delta\gamma_i$ was about 0.45, including the scatter due to intrinsic polarisation ($\sim$0.25) added in quadrature to measurement error (photon noise). Foreground galaxies were used to determine the intrinsic polarisation distribution. The smoothed shear field was determined for both the components of shear and these were used by the mass reconstruction programme along with appropriate co-efficients to generate the mass map.

While this method works quite well it has the disadvantage that the error weighting and the Gaussian smoothing are coupled to each other. Decreasing the Gaussian smoothing scale (to investigate finer structure in the mass map) reduces the effectiveness of the all important error weighting; in the limiting case when the Gaussian smoothing scale includes just one background source (on the average) there is no error weighting at all. Since lensing inversion is a highly non-linear process and the individual shear values were dominated by the errors on them, this resulted in the final reconstructed mass distributions being considerably dependent on the smoothing scale used. Often we could not discern any signal at all in the mass map when small smoothing scales were used.

To remedy this, apart from $\left<\gamma\right>$ we also calculated the error on it, $\delta{\left<\gamma\right>} = \Sigma(G_i/\delta\gamma_i)^2$/[ $\Sigma(G_i/(\delta\gamma_i)^2)$]² and used this to weight the shear values in the mass reconstruction algorithm. This removed the dependence of error weighting on the smoothing scale and made it possible to make mass maps with very small smoothing scales to (i) confirm that the lower resolution mass maps could be obtained by a post-reconstruction smoothing of the higher resolution map which indicated that our error-weighting and hence error estimates were correct, (ii) check the stability of the individual features seen in the mass reconstruction and (ii) investigate the mass distribution in better detail.

Source selection:

The source detection was done in the R-band using both IMCAT and SExtractor. Only those background (to the cluster) sources which were detected by IMCAT with a significance >7 and were also detected by SExtractor were used for the lensing analysis. Sources with neighbors closer than 5 pixels (1 arcsec) were eliminated to reduce the error in the shape estimation. With a series of subsamples spanning a narrow range of $\delta R = 0.5$ mag, we detected using Aperture Mass densitometry (described later) a lensing signal for sources in the range 22.5 < R < 26.5. This constituted our master list of 2550 sources for the mass reconstruction analysis. The sources from this master list which were detected in the other 3 bands ( $B \equiv 2080$, $V \equiv 2423$ and $I \equiv 2417$ sources) constituted the lensing analysis sample for those bands.

Quality of the mass reconstruction:

One of the advantages of using an FFT algorithm is that an output length scale is a natural feature and may be easily specified using certain parameters. This matching of the output resolution (of the mass-map) with the input shear field smoothing scale and data quality (for e.g., more wavemodes for better quality shear data and for higher resolution images) allowed us to make reconstuctions with different output resolutions to check the fidelity of mass features.

However, it has disadvantages as well. Any hole in the shear field (caused by a bright star, for example) is filled in with zeros by the algorithm leading to strong ripples and negative peaks in the reconstucted image. As a result one sometimes finds egregious artifacts which are 5-10 times larger than the noise. In general, the noise calculated over small regions (i.e. the "true" noise which avoids large scale correlated fluctuations) is 3-5 times smaller than an rms calculated over a large area including ripples and all. However, this latter quantity is perhaps more appropriate for determining the significance of the features in the mass maps and is the value listed in the figure captions. It must be noted that this is not the rms of a Gaussian random distribution and hence the usual rules of thumb and relationships of Gaussian distributions (e.g. >$3\sigma$ is significant) may not always be appropriate. We discuss below some of the ways, some heuristic and others more solid, in which we can deduce the reliability of the features seen in the mass reconstruction:
(i) The Curl-map: the shear field is a function of the gradient of the gravitational potential and so a mass map made from the curl of the shear field (effectively replacing $\gamma_1$ by $-\gamma_2$ and $\gamma_2$ by $\gamma_1$) must produce a structure-less noise map (Kaiser 1995) in the absence of systematic errors in the shear field. Thus, such a Curl-map may be expected to indicate the location and intensity of artifacts. Further, since the Curl-map is essentially free of source regions most of its pixels can be used to get a good estimate of the "large-scale'' rms discussed earlier.
(ii) Bootstrap techniques.
(iii) Compare mass maps made with different smoothing scales: features which vary from one scale to another in an inconsistent manner are likely to be artifacts.
(iv) Compare mass maps made with data from different bands: the shape of each lensed galaxy is approximately (but not exactly) the same in every band though the final shape should be different due to different PSFs and photon noise, especially for faint sources. It is necessary that a feature, in order to be considered real, should be of similar shape and intensity (within errors) in all the bands. Since the noise is so much stronger than the shear signal reproducing the same features in all the bands is an indication that PSF corrections and noise weighting were handled appropriately. Of course, this check is not sufficient to prove that the features are real since intrinsic ellipticities and locations of the background galaxies are similar/same in all the bands (see next point).
(v) Random shuffling of the Shear: the shear is sampled only at the positions of the background sources. Therefore, this multiplicative sampling function leaves its own convolved footprint on the mass map. However one can get a qualitative idea of the effect by keeping the positions fixed and randomly shuffling the observed shear values among them. Obviously this should again result in a structure-less noise map if there were no systematics introduced by the sampling function and the FFT.
(vi) An inspection of the location and significance (in terms of rms) of the negative peaks on the mass map itself.

Method 2

This method also used the raw IMCAT software from Kaiser's home page (see previous method) with some minor modifications for estimating the shear field. The mass reconstruction was done using the maximum likelihood estimator developed by Bartelmann et al. (1996) with the finite difference scheme described in Appendix B of Van Waerbeke et al. (1999). The reconstruction was not regularised and hence the resulting mass maps were noisier than those obtained from Method 1. However, as pointed out Van Waerbeke et al. (1999) and Van Waerbeke (2000) the advantage of method 2 is that the noise can be described analytically in the weak lensing approximation. When galaxy ellipticities are smoothed with a Gaussian window

$$W({{\textfont1=\bolditalics \hbox{$\bf\theta$ }}})={1\over \p... ...hbox{$\bf\theta$ }}}\vert^2\over \theta_{\rm c}^2}\right)\cdot$$ (1)

The noise in the reconstructed mass map is a 2-D Gaussian random field fully specified by the noise correlation function (Van Waerbeke 2000):

$$\langle N({{\textfont1=\bolditalics \hbox{$\bf\theta$ }}})N({... ...box{$\bf\theta$ }}}'\vert^2\over \theta_{\rm c}^2}\right)\cdot$$ (2)

We used this property to simulate shear fields (including the appropriate noise) to derive the positional stability of mass features and hence the significance of the offset between the mass centroid and the cD galaxy.

4.1.2 Distribution of the dark matter in MS 1008-1224

The mass reconstructions from B, V, R and I data using the first method are shown in Fig. 7 (30 arcsec smoothing scale) and Fig. 8 (15 arcsec smoothing scale).

Figure 7: Mass reconstruction of MS 1008-1224 from B, V, R and I images using the algorithm of Squires & Kaiser (1996) - Method 1 in the text - and a Gaussian smoothing of 30 arcsec. The iso-convergence contour interval is $\kappa = 0.025$. Broken contours represent negative values and the first solid contour delineates $\kappa =0$. The rms is about 0.025. The cross at (222, 222) marks the location of the cD galaxy. North is to the top and east is to the left.

Figure 8: Mass reconstruction of MS 1008-1224 from B, V, R and I images using the algorithm of Squires & Kaiser (1996) - Method 1 in the text - and a Gaussian smoothing of 15 arcsec. The bottom-left plot is the average of the 4 upper plots while the bottom-right is the average of the Curl-map in each band. The iso-convergence contour interval is $\kappa$ = 0.05. Broken contours represent negative values and the first solid contour delineates $\kappa =0$. The rms is about 0.05. The cross at (222, 222) marks the location of the cD galaxy. The concentric circles in the bottom-left plot, spaced 0 $.\mkern -4mu^\prime$4 apart, indicate the annuli used to determine the radial shear and mass profiles (Figs. 12 and 13); they are centred on the centroid of the mass distribution determined from the low resolution maps (Fig. 7). The two thick curved lines on the bottom-right plot trace the boundaries of the masks on the FORS1 images (to hide two very bright stars). North is to the top and east is to the left.

Features in the mass maps:

The lower resolution reconstructions (Fig. 7) are very similar in shape as well as magnitude of the peak ($\pm$5 per cent variation). The main features are a central mass condensation which seems to be elongated in the north-south direction, a fainter extension towards the north and a ridge leading off towards the north-east from the main component. The prominent cross at (222, 222) marks the location of the cD galaxy.

The higher resolution image (Fig. 8) clearly resolves the principal mass component into 2 peaks separated along (mostly) the north-south direction. Once again, it may be noted that the structures are similar in all the bands. We also note that the northern peak appears to be marginally higher than the southern one in all images. This montage (Fig. 8) also includes an average of the mass maps in the 4 bands (bottom-left). The concentric circles denote the annuli within which the mass was estimated to determine the radial profile. They are centred on the centroid of the mass distribution (the black dot) determined from the average of the lower resolution maps (Fig. 7). It must be noted that averaging the mass distribution from all the 4 bands does reduce the amplitude of spurious ripples relative to the more stable mass peaks but not by a factor of 2; for one, the noise is not Gaussian and for another, averaging the shear field with data from different bands reduces the photon noise but not the noise due to intrinsic galaxy ellipticity.

Also shown on the same plot is an average of the Curl-maps made in each of the 4 bands (bottom-right). There were two very bright stars in the MS 1008-1224 field which were masked on the FORS1 images. The thick curved lines delineate the extent of these masks from where no shear data was available. These holes in the shear field led to strong spurious peaks and ripples in the mass map.

Interpreting the mass features:

The two mass components at the centre are the most significant features in all the maps. Their stability across the different bands and smoothing scales as well as their high level of significance is a strong indicator that they are real features.

An inspection of the mass maps in Figs. 7 and 8 clearly shows that the strongest negative peak in each is in the region of the masks. Further, the north-eastern ridge mentioned earlier is along the boundary of one of the masks, its intensity varies from band to band (in contrast to the 2 principal mass components) and is very prominent in the Curl-map. So we concluded that it was a spurious feature spawned by the FFT and the masks. It was heartening to note that the strong spurious peaks generated by the masks were confined to their immediate vicinity and that much of the Curl-map, especially the lower half, mimics random noise with no strong features.

It is more difficult to determine if the faint but extensive signal leading to the north has a basis in reality. Given its faintness its considerable fluctuation from one map to another is only to be expected. But to a greater or lesser extent it is present in every single plot including others (not shown in this paper) obtained from different combinations of smoothing scale and wave-modes. So, we tentatively suggest that it is real but we shall not attempt to mine it for any further information. We only note that the cluster number and luminosity density distributions (Fig. 6) also show secondary peaks towards the north.

Another point that we shall discuss in more detail a little later is the offset between the cD galaxy and the centroid of the mass distribution.

Comparison with method 2:

The mass reconstructions obtained using method 2 are shown in Figs. 9 and 10 (20 and 15 arcsec smoothing scales, respectively).

Figure 9: Mass distribution of MS 1008-1224 using Method 2 and a Gaussian smoothing of 20 arcsec. It is superposed on the full R-band image obtained with the FORS1 on the VLT (440 arcsec on each side with north to the top and east to the left). The cD is the bright object located just below the central mass peak. The iso-convergence ($\kappa$) contour values are $\ldots -3\sigma, -2\sigma, -1\sigma, 1\sigma, 2\sigma, 3\sigma, \ldots$ (1 $\sigma = 0.05$).

Figure 10: High resolution mass reconstruction of MS 1008-1224 using Method 2 and a smoothing scale of 15 arcsec. The final plot was obtained by averaging the mass reconstructions in the 4 different bands. This averaging reduces the noise due to measurement error by a factor of 2 (but not that due to intrinsic ellipticity). The iso-convergence ($\kappa$) contour values are $\ldots -3\sigma, -2\sigma, -1\sigma, 1\sigma, 2\sigma, 3\sigma, \ldots$ (1 $\sigma = 0.07$). Each plot is 215 arcsec on a side with north to the top and east to the left. The crosswires mark the position of the cD galaxy. These maps are to be compared with the ones in Fig. 8 (method 1).

Clearly, the plot in Fig. 9, constructed using an intermediate smoothing scale, shows a resolution in between those seen in Figs. 7 and 8. The plots in Figs. 8 and 10 used the same resolution (but different algorithms) and indeed are very similar.

We compared the two methods quantitatively by carrying out a Pearson's r-test (Press et al. 1992) on the high resolution mass reconstructions (the Fig. 8 "Mass-av'' image of method 1 and Fig. 10 "I+R+B+V'' image of method 2). We obtained an r-coefficient of 0.837 for on-signal pixels and r = 0.298 for off-signal pixels. The smallest rectangle enclosing the 3$\sigma$ $\kappa$ contours of both images defined the on-signal region while the bottom quarter of the image was used for the off-signal region since this was the only clean area lacking the spurious features generated by the large masks in the upper half of the images (see Fig. 8 bottom-right plot). The on-signal correlation is very high and as expected much higher than the off-signal correlation. However the off-signal is still correlated because many galaxies are common in both methods (locations and shapes are the same), which will lead to correlated structures at the noise level throughout the map. Joffre et al. (2000) also measured a similar high correlation in the case of Abell 3667 even after masking the statistically significant regions of the mass distribution.

The Offset between the cD and the centroid of mass:

After analysing the data in many different ways, with different parameter sets and in different bands we concluded that the offset between the cD and the mass centroid was a real feature even though the magnitude of the offset varies from map to map. Qualitatively, the cD is to the south (and a little west) of the mass centroid in most mass plots. It is more likely to be associated with the southern (and less massive) mass component of the central double.

We used Method 2 to quantify the magnitude and significance of this offset. The best way for measuring the significance of the offset would have been to use an independent parametric model for the mass distribution and a parametric bootstrap method to generate a large number of mass reconstructions with different noise realisations and then measure the dispersion of the cD-mass centroid offset. Since such a model was not available we used the reconstructed mass map itself as the model. Combining different realisations of the noise (using the noise model of Eq. (2)), galaxy positions and intrinsic ellipticities we generated 5000 simulated datasets based on the I-band data at three different smoothing scales of 20, 30 and 40 arcsec. Figure 11 illustrates the positional stability one may expect in mass reconstructions and the numbers in Table 2 quantify the statistical significance of the offset.

Figure 11: Probability histograms of the location of the mass centroid obtained from parametric bootstrap resampling of the I-band shear data. The left hand plots correspond to the Y-axis offset and the right hand plots to the X-axis offset. The vertical dashed line indicates the position of the cD galaxy dashed line. The results have been plotted for 3 different smoothing scales. Note that in each plot the zero of the mass centroid is its average position; thus the cD is at different locations in each plot. Note also that the vertical scale is different for each plot (only the relative height of the bars within plot is relavant).

 

 

Table 2: The probabilities that the cD galaxy is offset from the mass centroid by more than 5 and 10 arcsec.

	$R_{\rm g}$ = 20''		$R_{\rm g}$ = 30''		$R_{\rm g}$ = 45''
	$\delta{X_{\rm c}}$	$\delta{Y_{\rm c}}$	$\delta{X_{\rm c}}$	$\delta{Y_{\rm c}}$	$\delta{X_{\rm c}}$	$\delta{Y_{\rm c}}$
Prob( $\delta > 5''$)	0.46	0.79	0.44	0.86	0.64	0.92
Prob( $\delta > 10''$)	0.21	0.48	0.20	0.68	0.30	0.73

Clearly, mass features move around on scales of $\sim$10 arcsec and this effect, naturally, increases with decrease of smoothing scale. One of the lessons we draw from this analysis is that squeezing finer mass details from the shear data is possible but at the expense of considerable flakiness in the positions of the features. It is important to reconstruct mass maps using various smoothing scales before attempting an interpretation of the same. Finally, the cD galaxy is offset to the south of the mass centroid by 19^+22.5_-18.5 arcsec (confidence level, CL = 90%); or, the cD is at least 5 arcsec (CL = 90%) south of the mass centroid.

4.1.3 Mass profile from tangential shear

The mass from weak shear, $M_{\rm shear}$, may be obtained from Aperture Mass Densitometry or the $\zeta$-statistics described by Fahlman et al. (1994) and Squires & Kaiser (1996). In brief, the average tangential shear in an annulus is a measure of the average density contrast between the annulus and the region interior to it; i.e. the average convergence $\kappa$ ($\equiv$ $\Sigma / \Sigma_{\rm cr}$), the ratio of the surface mass density to the critical surface mass density for lensing, as a function of the radial distance $\theta$ is given by

$$\kappa\left(<\theta_1\right)= \kappa\left(\theta_1<\theta<\theta_... ...{\theta_2} \left<\gamma_{\rm t}(\theta)\right> {\rm d} \left(\ln\theta\right) ,$$ (3)

where, $\gamma_{\rm t} = - (\gamma_1\cos{2\psi} + \gamma_2\sin{2\psi})$ is the tangential component of the shear with $\psi$ being the angle between the position vector of the object and the x-axis (RA-axis). The quantity $\left<\gamma_{\rm r}\right> = -\left<-\gamma_2\cos{2\psi} + \gamma_1\sin{2\psi}\right>$ is equivalent to the tangential component of the Curl of the shear field and is expected to be zero around any closed loop in the case of a shear field arising from gravitational lensing. Thus, the scatter in $\left<\gamma_{\rm r}\right>$is a measure of the the noise on $\left<\gamma_{\rm t}\right>$ due to intrinsic ellipticity, measurement noise and any other error, random or otherwise, in the shear field.

From Eq. (3), one can derive the average convergence within a series of apertures of radii $\theta_i$, $i = 1\ldots n$

$$\kappa\left(<\theta_i\right) = \kappa \left(\theta_n<\theta<\thet... ... \left<\gamma_{\rm t}\left(\theta\right)\right> {\rm d} \left(\ln\theta\right),$$ (4)

where, the region $[\theta_n, \theta_b]$ is the boundary annulus which provides the reference density (the first term) in excess of which the interior density values are obtained. Thus this method provides only a lower limit to the lensing mass estimate. It is to be noted that this expression has been cast such that the first term, which cannot be calculated and hence has to be neglected, is the average within the annulus [ $\theta_n, \theta_b$] and not the average density interior to $\theta_b$. Therefore, the effect of neglecting this term will be quite small if the data extend sufficiently far from the cluster centre. The mass within an aperture is given by

$$M (<\theta_i) = \kappa(<\theta_i) \ \Sigma_{\rm cr} \cdot \pi \le... ..._i^2 \,\frac{c^2}{4G} \left<\frac{D_{\rm ls}}{D_{\rm os}D_{\rm ol}}\right>^{-1}$$ (5)

where D is the angular size distance and its subscripts, l, o, and s, refer, respectively, to the lensing cluster ( z = 0.3062), observer (z = 0) and the background lensed sources ( $z = z_{\rm s}$). As has been explained earlier this FORS1+ISAAC dataset allowed us to estimate photometric redshifts of sources in the ISAAC field. We assumed that the 356 objects in the ISAAC field with 22.5 < R < 26.5 and good $z_{\rm phot}$ estimates (i.e. hyperz fits with $\chi ^2 < 1$) were representative of the FORS1 field as a whole. Of these, 302 lay behind the cluster at z > 0.31. From this redshift distribution we estimated $${ \left<\frac{D_{\rm ls}}{D_{\rm os}D_{\rm ol}}\right>^{-1}} = 1.38\pm0.03$$ Gpc ($1\sigma$ error). This value also includes a correction for the dilution of the lensing signal due to foreground objects being in the selected magnitude range. The mass is therefore given by

$$M(<\theta_i) = 6.14 \times 10^{14} ~h^{-1}~{M_{\odot}}\ \kappa(<\theta_i) \ {\theta'_i}^2 ,$$ (6)

where $\theta'_i$ is in arcmin. The fractional error on the mass estimate may be obtained by adding in quadrature the fractional errors in the distance modulus and $\kappa$. The latter is much larger but we may have underestimated the former. The uncertainty quoted for the distance modulus is simply the formal error on the mean of the distribution. To this must be added the (unknown) effect of stellar contamination at the fainter levels which would underestimate the mass (presumably small - see Griffiths 1994 and Conti 1999), biases in the photometric distribution (also small, as discussed previously), source clustering and cosmic variance. However, some of the effect of the latter two must have manifested in the noise on $\kappa$.

The radial profile of the shear is shown in Fig. 12.

Figure 12: Radial profile of shear in the MS 1008-1224 field. The filled circles are the tangential shear in successive annuli centered on the mass centroid (see Fig. 8). The open circles represent the Curlof the shear field which are expected to be (and are) distributed around zero if the shear field was due to gravitational lensing. The bars respresent $\pm 1\sigma$ errors.

The profile was determined in annuli centered on the mass centroid. The centre and the annuli, spaced 0 $.\mkern -4mu^\prime$4 apart, are marked on the bottom-left panel of Fig. 8 for reference. The annulus between 3 $.\mkern -4mu^\prime$2 and 3 $.\mkern -4mu^\prime$6 (560-635 h^-1 kpc from the mass centroid) was used as the boundary strip to set the zero of the density scale. The filled and the open circles represent the shear ( $\left<\gamma_{\rm t}\right>$) and the Curl of the shear field ( $\left<\gamma_{\rm r}\right>$), respectively. The $\left<\gamma_{\rm r}\right>$ values are indeed consistent with a zero value as expected for a lensing signal. Further, we confirmed that the error estimated for the individual shear values and used in the weighting was appropriate by checking that $\sqrt{\left<\gamma^2_{\rm r}\right>/N} \ \simeq (\Sigma^{^N}_{_1} 1/(\delta\gamma_{\rm r})^2)^{-1} \ \simeq (\Sigma^{^N}_{_1} 1/(\delta\gamma_{\rm t})^2)^{-1}$, where N is the total number of lensed galaxies in the shell.

The mass profile inferred from the shear is shown in Fig. 13 as a series of filled circles along with the error bars.

Figure 13: The mass profile from gravitational shear analysis in MS 1008-1224. The filled circles and the error bars ($1\sigma$) in each circle represents the inferred mass profile (from R-band data). The error bar is similar to the size of the circle in the first 2 data points. The superposed curves represent mass profiles of: top panel - Pseudo-Isothermal sphere (PIS) models with a velocity dispersion $\sigma _{\infty } = 925$ kms-1 and core radii of 0 $.\mkern -4mu^\prime$3, 0 $.\mkern -4mu^\prime$4 and 0 $.\mkern -4mu^\prime$5; middle panel: PIS models with $\sigma _{\infty } = 1000$ kms-1 and core radii of 0 $.\mkern -4mu^\prime$4, 0 $.\mkern -4mu^\prime$5 and 0 $.\mkern -4mu^\prime$6; bottom panel: NFW model with $r_{\rm s}=140 ~ h^{-1}$ kpc and $\delta _{\rm c}=3.52 \times 10^4$. The NFW curve is a fit to the data while the others are merely illustrative in nature. See the text (Sect. 5) for a discussion of these models.

It must be noted that the shear value (Fig. 12) does not fall to zero even at the largest radii which suggests that we may be underestimating the mass of this cluster. Also it must be noted that each data point in the mass profile contains the contribution from every other point farther out and so they are not independent.

4.2 Magnification bias and radial depletion

The combined effect of deflection and magnification of light, results in a modification of the number density of galaxies seen through the lensing cluster. In the case of a circular lens, the galaxy count at a radius $\theta$ is

$$N({<}m,\theta)=N_0(<m) \mu(\theta)^{2.5\alpha-1} ,$$ (7)

where $\alpha$ is the intrinsic slope (without lensing) of the galaxy counts, $\mu$ the gravitational magnification, and N₀ the intrinsic galaxy number density (hereafter, the zero-point). Depending on the value of $\alpha$, we may observe an increase or a decrease in the number of galaxies in the central region out to a limiting radius which depends on the shape of the gravitational potential and the redshift distribution of the background sources. The magnification bias has already been observed in some lensing clusters (Broadhurst et al. 1995; Fort et al. 1996; Taylor et al. 1998; Broadhurst 1998). It is particularly obvious in very deep observations as the slope of galaxy counts decreases to values as low as $\alpha \approx 0.2$ at faint levels.

4.2.1 Evidence of depletion of lensed sources

We considered as foreground those galaxies at $z_{\rm phot} < 0.25$ and as background (lensed) those at $z_{\rm phot} > 0.4$ (see Sect. 3.2). We minimized misclassification by considering only those which had a good photometric redshift solution (hyperz fit $\chi ^2 < 1$). To be consistent with the shear analysis, we considered only the $22.5 \le R \le 26.5$ (method 1) and $22.5 \le I \le 25.5$ (method 2) samples. The galaxy counts slopes for the 2 samples were found to be 0.192 and 0.233, respectively, suggesting that depletion would be significant in the FORS1 and ISAAC data.

Figure 14 shows the projected number density of galaxies having good photometric redshifts from BVRIJK data and in the magnitude range $22.5 \le I \le 25.5$.

Figure 14: Galaxy distribution in the ISAAC field (X and Y are in arcsec with north to the top and east to the left). Each point on the plot represents the location of a galaxy from the photometric redshift sample ( 22.5 < I < 25.5 and $\chi ^2 (z_{\rm phot}) < 1$)). The contours represent the galaxy number density field (smoothed with a 10'' Gaussian) in units of galarcmin-2. Foreground galaxies (z<0.25) are plotted in the top panel. The average number density is 8 gal arcmin-2. The bottom panel represents the background (lensed) galaxies (z>0.4). The density averaged over the field is 36 gal arcmin-2. The lowest density is close to the centre where it drops to 4 gal arcmin-2. The density peaks at 180 gal arcmin-2. The cross-wire marks the position of the cD galaxy. The distribution of foreground objects is, as expected, almost uniform. In contrast, the central depletion is clearly visible in the lensed galaxies. The centre of depletion is offset 7'' west and 9'' north of the cD. Also seen is the excess of galaxies in the south-west quadrant (quadrant Q4), which appears to be a cluster of galaxies at $z \sim 0.9$ lensed by MS 1008-1224.

Galaxies at z > 0.4 (background, lensed) are at the bottom. The control sample of foreground galaxies (z < 0.25) is plotted on top. In both cases, the density was corrected for the areas masked by bright galaxies. The foreground distribution is essentially random; in contrast, a strongly depleted area is visible near the centre of the field in the background distribution. This is the first evidence of the magnification bias effect based on a large sample of background galaxies with a known redshift distribution. The cross-wire marks the location of the cD galaxy. The offset of the cD ($\sim$10 arcsec south and 7 arcsec east) from the centre of depletion is consistent with the result from the shear analysis. The depletion and its offset from the cD are also seen in the R-selected sample.

4.2.2 Mass profile from magnification bias

The modification of the radial distribution of galaxy counts (i.e., the magnification bias) probes the amplitude of the projected mass density. In the weak lensing regime the relation simplifies to:

$$N({<}m,\theta) N_0({<}m) \mu(\theta)^{2.5\alpha-1}$$ = (8)

$$\approx N_0({<}m)\left[1+2\kappa\left(\theta\right)\right]^{2.5\alpha-1}.$$ (9)

The mass inside a radius $\theta$ (arcmin) is given by:

$$M(\theta)= 4.4 \times 10^{14}~ h^{-1}~ {M_{\odot} } \left<{D_{\rm... ...\over {\rm 1~Gpc}}\right) \int_0^{\theta} \theta \kappa(\theta) {\rm d}\theta .$$ (10)

The slope of the galaxy counts, $\alpha$, and the depletion curves, can be be estimated directly from the data. The quantity $\left<{D_{\rm ls}/ D_{\rm os}}\right>^{-1}$ may be computed from the redshift distribution shown in Fig. 4. So, in principle, one can get $\kappa(\theta)$ and hence the radial mass distribution by merely counting galaxies on the FORS1/ISAAC images but in practice we could not determine the zero-point, N₀, satisfactorily due to two reasons:

(i) The depletion extends beyond the ISAAC field.

(ii) A lensed background cluster: we detected a significant enhancement of galaxy number density on the bottom-right (hereafter Q₄) quadrant of the ISAAC field (Fig. 14, lower panel). A visual inspection of the FORS1 images showed faint and distorted galaxies between a radius of 50 and 80 arcsec from the centre of depletion. We compared the photometric redshift distribution of galaxies in Q₄ with that from the other three quadrants (Q_1-3). The difference between the (area-normalised) galaxy numbers in Q₄ and Q_1-3 are plotted in Fig. 15.

Figure 15: A comparison of the the redshift distribution of galaxies in the South-West quadrant (Q4) and the other 3 quadrants combined (Q1-3). The difference in the galaxy numbers have been plotted after normalising the numbers by the respective areas. An excess of galaxies is seen at redshift 0.9 at a $6\sigma$ significance level. We conclude that a distant cluster of galaxies lies at this location and redshift.

It shows a significant excess of galaxies at redshift 0.9. Quantitatively, in the magnitude range $22.5 \le I \le 25.5$, the number of galaxies at $z_{\rm phot} = 0.8 {-} 1.1$ is 61 in Q₄, whereas the three other quadrants together have 38 galaxies in the same redshift range, i.e., a $\sim$$6\sigma$ excess. We conclude therefore that there is a distant cluster of galaxies behind MS 1008-1224 at $z \approx 0.9$. Remarkably, the smoothed number density contours are distorted in a manner similar to the average shear pattern of galaxies at that location, as if the background cluster itself has been globally magnified and sheared. This is the first case of cluster-cluster lensing known. However, it has the unfortunate consequence of "spuriously" deepening the MS 1008-1224 radial depletion profile.

So we tried to determine the asymptotic zero-point for ISAAC by extrapolating from the FORS1 field as a whole. We selected galaxies from the FORS1 field which were fainter than the brightest cluster members and outside the cluster sequence on the Colour-Magnitude plot. We then computed the radial galaxy number density from the FORS1 data within the ISAAC area, excluding the the background cluster. Figure 16 shows the depletion curves for the FORS1 field as well as for the ISAAC subsamples with photometric redshifts.

Figure 16: Galaxy number density as function of the radial distance from the centre of depletion. All the curves are for the 22.5<I<25.5galaxy sample. The flat level in the FORS1 curve at large radius provided the zero-point for the full sample. The other curves were computed from galaxies inside the ISAAC field having a good photometric redshift. The thick dashed curves are straight line fits to the FORS1 and ISAAC samples in the inner regions and were used to calculate the fraction of ISAAC lensed galaxies with a good photometric redshift (see text).

A central depletion as well as a flat distribution at large distance ( $N_0^{\rm FORS}$, which includes all $22.5 \le I \le 25.5$ galaxies, irrespective of redshift) beyond the extent of the ISAAC field are visible.

$$N^{\rm FORS} = N_0^{{\rm FORS}(z>0.4)}\mu^{2.5\alpha-1} +F_{\rm fg}$$ (11)

where $F_{\rm fg}$ is the density of unlensed galaxies (foreground galaxies plus perhaps a few cluster members) and $N_0^{{\rm FORS}(z>0.4)}$ is the zero point of the lensed background galaxies at z>0.4 in the FORS1 field. A rough estimate of $F_{\rm fg} \approx 9$ gal arcmin^-2 is obtained from the intercept of the FORS1 radial curve on the vertical axis. At large distances, the magnification is negligible, i.e. $\mu = 1$, and $N^{\rm FORS}$ is the same as $N_0^{\rm FORS}$. Using these in Eq. (11), we obtain $N_0^{{\rm FORS}(z>0.4)} = 40\pm6.5$ gal arcmin^-2 (CL = 90%)

For the $z_{\rm phot} > 0.4$ sample in the ISAAC field we have

$$N^{{\rm ISAAC}(z_{\rm p}>0.4)} = N_0^{{\rm ISAAC}(z_{\rm p}>0.4)}\mu^{2.5\alpha-1} .$$ (12)

But

$$N_0^{{\rm ISAAC}(z_{\rm p}>0.4)} = k\ N_0^{{\rm FORS}(z>0.4)}$$ (13)

where k is the (unknown) fraction of z > 0.4 galaxies which have been identified as such using photometric redshifts. So k is the key to determining mass from the depletion analysis. From Eqs. (11) and (13) we find

$$k={N^{{\rm ISAAC}(z>0.4)} \over N^{\rm FORS} -F_{\rm fg}} \cdot$$ (14)

k is simply the ratio of the depletion curves for the (foreground subtracted) FORS1 and $z_{\rm phot}$ samples. This ratio fluctuates considerably due to Poisson noise and clustering. As a first approximation we fit straight lines to segments of the depletion curves inside 0 $.\mkern -4mu^\prime$67 (to exclude the background cluster) and derived an average value of ${<k>} = 0.65\pm0.17$, a zero-point of $26\pm8$ gal arcmin^-2 for the $z_{\rm phot} > 0.4$ sample and finally a rather uncertain mass estimate of $6.6\pm2.2 \times 10^{14} ~h^{-1} \,M_{\odot}$ (CL = 90% for all) within 1 arcmin radius which is much larger than the X-ray and weak shear estimates.

In conclusion, despite the very good data set this method is still saddled with large errors. Depletion analysis is in principle very simple (just counting galaxies) and a neat way of avoiding the mass sheet degeneracy of shear analysis but is plagued by cosmic variance, background clustering and, as we have demonstrated here, the necessity of having complete redshift samples.