3 $\vec{M}$ $_{{\rm ap}}$ statistical analysis

Over the whole field of view of our $50'' \times 50''$ STIS image, we recognize a strong apparent tangential alignment of galaxies towards the center (see Figs. 1 and 2). Although the small $50'' \times 50''$ field does not allow a quantitative weak lensing analysis to verify or to falsify the hypothesis of a massive dark matter halo in the image center, we use the $M_{\rm ap}$ statistic (Schneider 1996) to test whether we can identify a significant and stable peak. The details of the creation of an object catalog are the same as for the rest of the SPSS project. They are described and justified in depth in Hämmerle et al. (2002). In short, we performed an analysis of the data using two different software packages. Objects were detected with SExtractor and a modified version of the IMCAT package (Kaiser et al. 1995 hereafter referred to as KSB; Erben et al. 2001). Finally, we only retained objects detected with both programs, assigning them the photometric information from SExtractor and the shape parameters from IMCAT. There are not enough foreground stars in a STIS image to allow directly a PSF correction in the KSB style from the data itself, which is the usual procedure with wide-field ground-based observations. On the other hand, our analysis of STIS data, especially starfields obtained over the same period of time as our observations, showed that the PSF is sufficiently stable (Hämmerle et al. 2002) so that these starfields provide a good estimate of the quantities used for the KSB anisotropy and smearing correction (see also Hoekstra et al. 1998 for a similar procedure with WFPC2 data). The final correction was performed using the PSF information from the starfields as described in Erben et al. (2001). We removed from the catalog the objects with an axis ratio larger than two; those will be considered in the next section as candidate arclets. Their alignment towards the direction defined by the $M_{\rm ap}$ statistics will provide an independent test of the shear signal. In the end, our catalog contained 54 objects, with fully corrected ellipticities, which were used for the subsequent analysis.

Figure 1: The figure shows contour maps of the S/N of $M_{\rm ap}$ values (Eq. (2)) superimposed on the field. The S/N ratio was calculated on a $20\times 20$ grid of our image. The filter scale $\theta$ is 25''. White contours correspond to S/N=1.0, 1.5, 2.0 and black contours to S/N=2.1, 2.2, 2.3. See the text for a discussion on the robustness of this result.

We use standard lensing notations in this paper. For a broader introduction to this topic, the reader should refer to Bartelmann & Schneider (2001). We used the $M_{\rm ap}$ statistic (Schneider 1996) for the weak lensing analysis. The family of $M_{\rm ap}$ statistics uses the fact that a filtered integral over the convergence $\kappa$ can be converted into a filtered integral over the observable tangential shear $\gamma_{\rm t}$ (see Eqs. (9) and (15) in Schneider 1996), if the filter function $U(\vert{\mbox{\boldmath$\vartheta$ }}\vert)$satisfies $\int_0^\theta{\rm d}\theta'\;\theta'~U(\theta')=0$ but is arbitrary otherwise. It is straightforward to construct an unbiased estimate $M_{\rm ap}'$ for the integral by a discrete sum over observed image ellipticities $\epsilon_{{\rm t}}$ and considering the coordinates origin being at the center:

$$M_{\rm ap}'(\theta)={\pi\theta^2\over N}\sum_i \epsilon_{\rm ... ...rtheta$ }}_i)~ Q(\vert{\mbox{\boldmath$\vartheta$ }}_i\vert)\;$$ (1)

with $Q(x)=q(x/\theta)/\theta^2$ and $q(\rho)=(6/\pi)\rho^2(1-\rho^2)$.

Moreover, as it is a scalar quantity, expressions for the variance and hence the S/N for a derived $M_{\rm ap}$ value are easily calculated. For the S/N one obtains (see Schneider 1996):

$$S={\sqrt{2}\over \sigma_\epsilon} {\sum_i {\epsilon_{\rm t}(... ...2\left(\vert {\mbox{\boldmath$\vartheta$ }}_i\vert\right)}}},$$ (2)

where $\sigma_{\epsilon}$ is the intrinsic dispersion of image ellipticities. It is estimated by the dispersion of the galaxies entering the calculation of S [ $\sigma_{\epsilon}^2=N^{-1}\sum_i (\epsilon_{{1}i}^2+\epsilon_{{2}i}^2$)]. Hence the $M_{\rm ap}$statistics are the ideal tool to evaluate the significance and robustness of a mass concentration. In our case, with only a $50'' \times 50''$ field and 54 objects in hand, the possibilities for this kind of analysis are limited. We are confined to small radii for our filter scale $\theta$ (about 25'') for which the $M_{\rm ap}$statistic is extremely noisy. Moreover, applying (1) to a grid of points on our data will not give an unbiased estimate of $M_{\rm ap}$ since the data do not cover a complete circle with radius $\theta$ at most positions. However, even if $M'_{\rm ap}$ can no longer be interpreted as an unbiased estimate of the filtered density profile, it yields a measure of the tangential alignment around a point and as such is a useful statistical quantity.

The application of this approach to our data gives the results shown in Fig. 1. There, we show a significance map for one filter radius ( $\theta = 25''$). We could slightly vary the size of the filter, but we are limited by the size of the field and the number of objects in it. Changing the filter size down to 20'' or up to 30'' does not modify the results. We recover a peak located at the image pixel coordinates x=1066; y=1297 (here and in the following, "pixels'' denote the subsampled pixels of size 0.025'' and with an origin for the coordinate system at the lower left corner of the image), whose position approximately coincides with the position of the brightest galaxy in the field. With a $S/N\approx 2.42$ for this smoothing, the peak is marginally significant. Here, the significance was estimated with (2). We found that these estimates are in very good agreement with those obtained from randomizing the orientations of galaxy ellipticities. To judge whether the peak is a robust statistical signal, we repeated the calculation of the $M_{\rm ap}$statistics drawing randomly $70\%$ and $50\%$ of the galaxies out of the initial 54 objects catalog. This exercise was repeated 100 times and the positions of the most significant peaks in the realizations were considered. The mean and standard deviation for the recovered positions of the peak are in the $70\%$ case: $\left\langle x \right\rangle=1250; \sigma_x=265; \left\langle y \right\rangle=1383; \sigma_y=244$ and for the $50\%$ case: $\left\langle x \right\rangle=1234; \sigma_x=300; \left\langle y \right\rangle=1286; \sigma_y=350$. The number of "catastrophic outliers'', where the most significant peak is located at the very border of the frame, is about $12\%$ in both cases. The mean amplitude for the peaks is $S/N\approx 2.4$ ($70\%$ case) and $S/N\approx 2.1$ ($50\%$ case). These values are a bit higher than expected from simple Gaussian statistics. Also if we choose our samples not randomly but remove the $30\%$ and $50\%$ most elliptical galaxies from the initial catalog, we recover peaks consistent in position with these results (although in the $30\%$ case, a strong peak appears at the border of the frame).

Finally we checked the probability of obtaining at least a $S/N\approx 2.42$ detection in fields with randomly oriented galaxy ellipticities. As mentioned above, Eqs. (1) and (2) applied to our field do not provide unbiased estimates of $M_{\rm ap}$ and its uncertainty, since our data do not cover a complete circle of the smoothing radius at the peak position. In order to estimate this probability, we generated 10 000 $50'' \times 50''$ fields with 54 objects randomly placed, with random galaxy ellipticities drawn from the distribution $P(\epsilon_1, \epsilon_2)\propto \exp(-\vert\epsilon\vert^2/ \sigma_{\epsilon}^2)$ with $\sigma_{\epsilon}=0.36$ matching the $\sigma_{\epsilon}$ of our data. We find that we can reproduce a peak of at least $S/N\approx 2.42$ with $700<x_{\rm peak}<1300; 1000<y_{\rm peak}<1600$ in $2.85\%$of all cases. We conclude that the 54 objects detected in our frame show a fairly robust alignment towards the upper middle part of our image. As stated above, the lack of knowledge about the shear outside the Slens1 field does not permit decisive conclusions about the existence of a possible massive structure within the field.