3. Galaxy shape analysis
The galaxies have been processed using the IMCAT software generously made available by Nick Kaiser 4. Some of the process steps have been modified from the original IMCAT version in order to comply with our specific needs. These modifications are described now.
The object detection, centroid, size and magnitude measurements are done using SExtractor (Bertin & Arnouts 1996 5) which is optimized for the detection of galaxies. We replace the parameter (physical size of an object), calculated in the IMCAT peak finder algorithm by the half-light-radius of SExtractor (which is very similar to measured in IMCAT). This lowers the signal-to-noise of the shape measurements slightly, but it is not a serious issue for the statistical detection of cosmic shear described in this work. Before going into the details of the shape analysis, we first briefly review how IMCAT measures shapes and corrects the stellar anisotropy. Technical details and proofs can be found in Kaiser et al. 1995 (hereafter KSB), Hoekstra et al. 1998 and Bartelmann & Schneider 1999a.
3.1. PSF correction: the principle
KSB derived how a gravitational shear and an anisotropic PSF affect the shape of a galaxy. Their derivation is a first order effect calculation, which has the nice property to separate the gravitational shear and the atmospheric effects. The correction is calculated first on the second moments of a galaxy, and subsequently the galaxy ellipticity can be directly expressed as a function of the shear and the star anisotropy. The raw ellipticity of an object is the quantity measured from the second moments of the surface brightness :
The aim of the window function is to suppress the photon noise which dominates the objects profile at large radii. According to KSB, in the presence of a shear and a PSF anisotropy , the raw ellipticity is sheared and smeared, and modified by the quantity :
The shear and smear polarization tensors and are measured from the data, and the stellar ellipticity , also measured from the data, is given by the raw stellar ellipticity :
Using Eq. (2) and Eq. (3) we can therefore correct for the stellar anisotropy, and obtain an unbiased estimate of the orientation of the shear . To get the right amplitude of the shear, a piece is still missing: the isotropic correction, caused by the filter and the isotropic part of the PSF, which tend to circularize the objects. Luppino & Kaiser 1997 absorbed this isotropic correction by replacing the shear polarization in Eq(2) (which is an exact derivation in the case of a Gaussian PSF) by the pre-seeing shear polarisability :
This factor `rescales' the galaxy ellipticity to its true value without changing its orientation, after the stellar anisotropy term was removed. The residual anisotropy left afterwards is the cosmic shear , therefore the observed ellipticity can be written as the sum of a `source' ellipticity, a gravitational shear term, and a stellar anisotropy contribution:
There is no reason that should be the true source ellipticity , as demonstrated by Bartelmann & Schneider 1999a. The only thing we know about is that implies . Therefore Eq. (5) provides an unbiased estimate of the shear as long as the intrinsic ellipticities of the galaxies are uncorrelated (which leads to ). The estimate of the shear is simply given by
The quantities , and are calculated for each object. The shear estimate per galaxy (Eq(6)) is done using the matrices of the different polarization tensors, and not their traces (which corresponds to a scalar correction) as often done in the literature. Although the difference between tensor and scalar correction is small (because is nearly proportional to the identity matrix), we show elsewhere, in a comprehensive simulation paper (Erben et al. 2000), that the tensor correction gives slightly better results.
3.2. PSF correction: the method
The process of galaxy detection and shape correction can be done automatically, provided we first have a sample of stars representative of the PSF. However, in practice the star selection needs careful attention and cannot be automated because of contaminations. Stars can have very close neighbor(s) (for instance a small galaxy exactly aligned with it) that their shape parameters are strongly affected. Therefore we adopted a slow but well-controlled manual star selection process: on each CCD, the stars are first selected in the stellar branch of the diagram in order to be certain to eliminate saturated and very faint stars. We then perform a clipping on the corrected star ellipticities, which removes most of the stars whose shape is affected by neighbors (they behave as outliers compared to the surounding stars). It is worth noting that the clipping should be done on the corrected ellipticities and not on the raw ellipticities, since only the corrected ellipticities are supposed to have a vanishing anisotropy. The stellar outliers which survived the clipping are checked by eye individually to make sure that no unusual systematics are present.
During this procedure, we also manually mask the regions of the CCD which could potentially produce artificial signal. This includes for example the areas with very strong gradient of the sky background, like around bright stars or bright/extended galaxies, but also spikes produced along the diffraction image of the spider supporting the secondary mirror, columns containing light from saturated stars, CCD columns with bad charge transfer efficiency, residuals from transient events like asteroids which cross the CCD during the exposure and finally all the boundaries of each CCD. At the end, we are left with a raw galaxy catalogue and a star catalogue free of spurious objects, and each CCD chip has been checked individually. This masking process removes about 15% of the CCD area and the selection itself leaves about 30 to 100 usable stars per CCD.
The most difficult step in the PSF correction is Eq. (6), where the inverse of a noisy matrix is involved. If we do not pay attention to this problem, we obtain corrected ellipticities which can be very large and/or negative, which would force us to apply severe cuts on the final catalogue to remove aberrant corrections, thus losing many objects. A natural way to solve the problem is to smooth the matrix before it is inverted. In principle should be smoothed in the largest possible parameter space defining the objects: might depend on the magnitude, the ellipticity, the profile, the size, etc... In practice, it is common to smooth according to the magnitude and the size (see for instance Kaiser et al. 1998, Hoekstra et al. 1998). Smoothing performed on a regular grid is generally not optimal, and instead, we calculate the smoothed for each galaxy from its nearest neighbors in the objects parameter space (this has the advantage of finding locally the optimal mesh size for grid smoothing). Increasing the parameter space for smoothing does not lead to significant improvement in the correction (which is confirmed by our simulations in Erben et al. 2000), therefore we keep the magnitude and the size to be the main functional dependencies of .
A smoothed does not eliminate all abnormal ellipticities; the next step is to weight the galaxies according to the noise level of the ellipticity correction. Again, this can be done in the gridded magnitude/size parameter space where each cell contains a fixed number of objects (the nearest neighbors method). We then calculate the variance of the ellipticity of those galaxies, which gives an indication of the dispersion of the ellipticities of the objects in the cell: the larger , the larger the noise. We then calculate a weight w for each galaxy, which is directly given by :
where is a free parameter, which is chosen to be the maximum of the ellipticity distribution of the galaxies. Eq. (7) might seem arbitrary compared to the usual weighting, but the inverse square weighting tends to diverge for low-noise objects (because such objects have a small ), which create a strong unbalance among low noise objects. The aim of the exponential cut-off as defined in Eq. (7) is to suppress this divergence 6.
The weighting function prevents the use of an arbitrary and sharp cut to remove the bad objects. However, we found in our simulations (Erben et al. 2000) that we should remove objects smaller than the seeing size, since they carry very little lensing information, and the PSF convolution is likely to dominate the shear amplitude. Our final catalogue contains about 191000 galaxies, of which 23000 are masked. It is a galaxy number density of about , although the effective number density when the weighting is considered should be much less. We find , which corresponds to the ellipticity variance of the whole catalogue.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000