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Astron. Astrophys. 358, 30-44 (2000)

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5. Analysis of the systematics

Now we have to check that the known systematics cannot be responsible for the signal. In the following we discuss three types of systematics:

  • The intrinsic alignment of galaxies which could exists in addition to the lensing effect. We assume such an alignment do not exist, but the overlapping isophotes of close galaxies produces it. We could in principle remove this effect by choosing a window function small compared to the galaxy distance in the pair, such that close galaxies do not influence the second moment calculation of themselves. However this is difficult to achieve in practice.

  • The strongest known systematics is the PSF anisotropy caused by telescope tracking errors, the optical distortion, or any imaginable source of anisotropy of the star ellipticity. We have to be sure that the PSF correction outlined in Sect. 2.2 removes any correlation between galaxy and star ellipticities.

  • The spurious alignment of galaxies along the CCD frame lines/columns. We cannot reject this possibility since charge transfer along the readout directions is done by moving the charges from one pixel to the next pixel and so forth, with an transfer efficiency of 0.99998. This effect could spread the charges of the bright objects (very bright and saturated stars produce this kind of alignment, but they have been removed during the masking procedure). Therefore we can expect the objects to be elongated along the readout direction.

5.1. Systematics due to overlapping isophotes

Let us consider the first point in the above list of systematics. In order to study the effect of close galaxy pairs, we measured the signal by removing close pairs by varying a cut-off applied on the respective distance of close galaxies. Fig. 3 shows the signal measured when successively closer pairs with [FORMULA] (no pair rejection), 5, 10 and 20 pixels have been rejected. The cases [FORMULA] and [FORMULA] show an excess of power at small scales compared to [FORMULA] and [FORMULA] (the latter two give the same signal). Therefore we assume that for [FORMULA] we have suppressed the overlapping isophote problem, and in the following we keep the [FORMULA] distance cut-off, which gives us a total of [FORMULA] galaxies for the whole data sets, as already indicated at the end of Sect. 3.2. By removing close pairs of galaxies, we also remove the effect of possible alignment of the ellipticities of galaxies in a group due to tidal forces.

[FIGURE] Fig. 3. The thick solid line shows the signal as plotted on Fig. 1. It was obtained with a catalogue of galaxies where galaxies closer than 10 pixels were rejected. The three other curves show the same signal measured with different rejection criteria: the diamond-dotted line is for no rejection at all, the triangle-dashed line for galaxies closer than 5 pixels rejected, and the square-dashed line for galaxies closer than 20 pixels rejected. This figure illustrates that the overlapping isophotes of close galaxies tends to overestimate the shear.

5.2. Systematics due to the anisotropic PSF correction

We next study the second point concerning the residual of the PSF correction. Fig. 11 shows that the star ellipticity correction is efficient in removing PSF anisotropies. The raw star ellipticity can be as large as [FORMULA] in the most extreme cases. Figs. 12 to 19 show maps of the uncorrected and the corrected star ellipticities. The same camera used at different times clearly demonstrates that the PSF structure can vary a lot in both amplitude and orientation, and that it is not dominated by the optical distortion (as we can see from the location of the optical center, given by the dashed cross). Individual CCD's are [FORMULA] chips, easily identified by the discontinuities in the stellar ellipticity fields.

Next, let us sort the galaxies according to the increasing stellar ellipticity, and bin this galaxy catalogue such that each bin contains a large number of galaxies. We then measure, for each galaxy bin, two different averaged galaxy ellipticities [FORMULA]: one is given by Eq. (5) and the other by Eq. (5), without the anisotropy correction [FORMULA]. The former should be uncorrelated with the star ellipticity if the PSF correction is correct (let us call [FORMULA] this average); and the latter should be strongly correlated with the star ellipticity, (let us call [FORMULA] this average). Since the galaxies are binned according to the stellar ellipticity, galaxies of a given bin are taken from everywhere in the survey, therefore the cosmic shear signal should vanish, and the remaining possible non vanishing value for [FORMULA] and [FORMULA] should be attributed to a residual of star anisotropy. Fig. 4 shows [FORMULA] and [FORMULA] (dashed lines) and [FORMULA] and [FORMULA] (solid lines) versus respectively [FORMULA] and [FORMULA]. The solid lines exhibit a direct correlation between the galaxy and the star ellipticities, showing that the PSF anisotropy does indeed induce a strong spurious anisotropy in the galaxy shapes of a few percents. However, the dashed lines show that the corrected galaxy ellipticities are no longer correlated with the star ellipticity, the average [FORMULA] fluctuates around [FORMULA], while [FORMULA] is consistent with zero. This figure shows the remarkable accuracy of the PSF correction method given in KSB. Error bars in these plots are calculated assuming Gaussian errors for the galaxies in a given bin. The significant offset of [FORMULA] of [FORMULA] might be interpreted as a systematic induced by the CCD, as we will see in the next section, and can be easily corrected for. Fig. 5 shows that this systematic is nearly galaxy independent, and affect all galaxies in the same way. This is also in favor of the CCD-induced systematic, since we expect that a PSF-induced systematic (which is a convolution) would depend on the galaxy size.

[FIGURE] Fig. 4. Average galaxy ellipticity [FORMULA] versus the average star ellipticity [FORMULA] for both components [FORMULA]. The dashed lines are obtained from the fully corrected galaxy ellipticities, as given by Eq. (5). The solid lines are obtained from the galaxy ellipticities corrected from the seeing, but without the anisotropy correction term [FORMULA] of Eq. (5). Each ellipticity bin contains about [FORMULA] galaxies, and the error bars are calculated assuming Gaussian errors [FORMULA]. Except for a constant tiny bias along the [FORMULA] direction, the corrected galaxies are uncorrelated with the stellar ellipticity, which demonstrates that the PSF correction method works well.

[FIGURE] Fig. 5. Average galaxy ellipticity [FORMULA] (solid line) and [FORMULA] (dashed line) as a function of the object size [FORMULA]. It is shown that the systematic bias of [FORMULA] along the [FORMULA] component is fairly galaxy independent.

Fig. 6 shows the same kind of analysis, but instead of sorting the galaxies according to the star ellipticity amplitude, galaxies are now sorted according to the distance r from the optical center. The average quantities we measure are no longer [FORMULA] and [FORMULA] versus [FORMULA] and [FORMULA], but the tangential and the radial ellipticity [FORMULA] and [FORMULA] versus r. This new average is powerful for extracting any systematic associated with the optical distortion. Fig. 6 shows that the systematics caused by the optical distortion are a negligible part of the anisotropy of the PSF, as we should expect from Figs. 12 to 19 (where the PSF anisotropy clearly does not follow the optical distortion pattern).

[FIGURE] Fig. 6. Average tangential galaxy ellipticity [FORMULA] and radial galaxy ellipticity [FORMULA] versus the distance from the optical center r. As for Fig. 4, the dashed lines are obtained from the fully corrected galaxy ellipticities, as given by Eq. (5), and the solid lines are obtained from galaxy ellipticities corrected from seeing but where the anisotropy correction term [FORMULA] has been removed from Eq. (5). Each ellipticity bin contains about [FORMULA] galaxies, and the error bars are calculated assuming Gaussian errors [FORMULA]. The absence of a significant amplitude between the dashed and the solid lines show that the optical distortion effect is a negligible contribution to the PSF anisotropy.

5.3. Systematics due to the CCD frames

Using the same method as in the previous section, we can also investigate the systematics associated with the CCD line/columns orientations. Here, instead of sorting the galaxies according to the star ellipticity or the distance from the optical center, the galaxies are sorted according to their X or Y location on each CCD frame. By averaging the galaxy ellipticities [FORMULA] and [FORMULA] in either X or Y bins, we also suppress the cosmic shear signal and keep only the systematics associated with the CCD frame. Fig. 7 shows [FORMULA] and [FORMULA] (dashed lines) and [FORMULA] and [FORMULA] (solid lines) versus [FORMULA] and [FORMULA]. The plots from the top-left to bottom-right correspond respectively to [FORMULA] versus [FORMULA], [FORMULA] versus [FORMULA], [FORMULA] versus [FORMULA], and [FORMULA] versus [FORMULA]. We see that [FORMULA] is systematically negative by [FORMULA] for both X and Y binnings, while [FORMULA] does not show any significant deviation from zero. This result is fully consistent with the dashed lines in Fig. 4 which demonstrate that the [FORMULA] systematic is probably a constant systematic which affects all the galaxies in the same way, and which is not related to the star anisotropy correction. The origin of this constant shift is still not clear, it might have been produced during the readout process, since a negative [FORMULA] corresponds to an anisotropy along columns of the CCDs.

[FIGURE] Fig. 7. Average galaxy ellipticity [FORMULA] versus the X and Y location on the CCDs. As for Fig. 4, the dashed lines are obtained from the fully corrected galaxy ellipticities, as given by Eq. (5), and the solid lines are obtained from the corrected galaxy ellipticities where the anisotropy correction term [FORMULA] has been removed from Eq. (5). The systematic negative mean value of [FORMULA] along lines or columns of the CCD (the two left panels) show that the galaxies are preferentially aligned with the columns of the CCD in the whole survey. A positive systematic value for [FORMULA] (the two right panels) is also visible, although much less significant.

5.4. Test of the systematics residuals

The correction of the constant shift of [FORMULA] along [FORMULA] has been applied to the galaxy catalogue from the beginning. It ensures that there is no more significant residual systematic (either star anisotropy or optical distortion or CCD frame), and demonstrates that the average level of residual systematics is small and much below the signal. However we have to check that the systematics do not oscillate strongly around this small value. If it were the case, then this small level of systematics could still contribute significantly to the variance of the shear. This can be tested by calculating the variance of the shear in bins much smaller than those used in Fig. 4 to calculate the average level of residual systematics. In order to decide how small the bins should be we can use the number of galaxies available in the apertures used to measure the signal, for a given smoothing scale. For example for [FORMULA] there is 45 galaxies in average, and for [FORMULA] there is 1100 galaxies. We can therefore translate a bin size into a smoothing scale, via the mean number of galaxies in the aperture. We found that the variance of the shear measured in these smaller bins is still negligible with respect to the signal, as shown by Fig. 8. The three panels from top to bottom show respectively the star anisotropy case, the optical distortion case and the CCD frame case. On each panel, the thick solid line is the signal with its error bars derived from 1000 randomizations. The short dashed lines show the [FORMULA] of these error bars centered on zero. On the top panel the two thin solid lines show [FORMULA] respectively measured with the galaxies sorted according to [FORMULA] and to [FORMULA]. The thin solid line in the middle panel shows [FORMULA] measured from the galaxies, sorted according to their distance from the optical center, and the two thin solid lines in the bottom panel show [FORMULA] measured on the galaxies sorted according to X and Y.

[FIGURE] Fig. 8a-c. Possible contribution of the systematics studied in Sect. 3.2 to the signal. On each of the plots, the thick solid line shows the signal as displayed in Fig. 1, and the dashed lines show the [FORMULA] fluctuation obtained from 1000 random realizations. From top to bottom: a  The two thin solid lines are [FORMULA] measured on the galaxies sorted according to the star ellipticity strength (see Fig. 4). For the different smoothing scales, the mean number and the variance of the number of galaxies in the chosen bins fit the one observed in the signal (thick solid) curve. b  the thin solid line is [FORMULA] measured on the galaxies sorted according to the distance from the optical center, and on c  the two thin solid lines correspond to the galaxies sorted according to their X or Y location on the CCDs.

In all the cases, the thin solid lines are consistent with the [FORMULA] fluctuation, without showing a significant tendency for a positive [FORMULA]. We conclude that the residual systematics are unable to explain the measured [FORMULA] in our survey, and therefore our signal is likely to be of cosmological origin.

A direct test of its cosmological origin is to measure the correlation functions [FORMULA], [FORMULA] and [FORMULA], where [FORMULA] and [FORMULA] are the tangential and radial component of the shear respectively:

[EQUATION]

where [FORMULA] is the position angle of a galaxy. If the signal is due to gravitational shear, we can show (Kaiser 1992) that [FORMULA] should be positive, [FORMULA] should show a sign inversion at intermediate scales, and [FORMULA] should be zero. This is a consequence of the scalar origin of the gravitational lensing effect and of the fact that galaxy ellipticity components are uncorrelated. Although we do not yet have enough data to perform an accurate measurement of these correlation functions, it is interesting to check their general behavior. Fig. 9 shows that in our data set, although the measurement is very noisy, both [FORMULA] and [FORMULA] are positive valued, while [FORMULA] is consistent with zero. This measurement demonstrates that the component of the galaxy ellipticities [FORMULA] of well separated galaxies are uncorrelated, and it is in some sense a strong indication that our signal at small scales is of cosmological origin.

[FIGURE] Fig. 9. From top to bottom, measurement of the correlation functions [FORMULA], [FORMULA] and [FORMULA]. The error bars are computed from 50 random realizations of our data set where the orientations of the galaxies were randomized.

The last thing we have checked is the stability of the results with respect to the field selection. We verified that removing one of the fields consecutively for all the fields (see Sect. 2 for the list of the fields) does not change the amplitude and the shape of the signal, even for the Abell 1942 field. The cluster has no impact and does not bias the analysis because it was significantly offset from the optical axis. This ensures that the signal is not produced by one field only, and that they are all equivalents in terms of image quality, PSF correction accuracy and signal amplitude, even using V and I colors. It also validates the different pre-reduction methods used for the different fields.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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