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

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6. Cosmological constraints

Fig. 1 provides a first comparison of our signal with some cosmological models. In order to rule out models we need to estimate first the sample variance in the variance of the shear. Although it has not been yet exactly derived analytically (because calculations in the non-linear regime are difficult), ray-tracing simulations can give an accurate estimate of it. We used the ray-tracing simulations of Jain et al. 1999 for this purpose.

Table 2 shows the two simulations we used. The [FORMULA]CDM model with [FORMULA] is not an independent simulation, but was constructed from the [FORMULA]CDM model with [FORMULA] simply by dividing [FORMULA] by 0.6. This should empirically mimic a model with both [FORMULA] and [FORMULA] equal to one. The redshift of the sources is equal to 1, which is not appropriate for our data. However, for the depth of the survey, we believe that it represents fairly well the mean redshift of the galaxies, which is the dominant factor in determining the second moment. Fig. 10 shows the amplitude and the scale dependence of the variance of the shear for the three cosmological models, compared to our signal. It is remarkable that models (1) and (3) can be marginally rejected (We did not plot the error bars due to the intrinsic ellipticity for clarity: they can be obtained from Fig. 8).

[FIGURE] Fig. 10. Comparison of our signal (thick line) with three cosmological models. The error bars are the cosmic variance measured on five independent realizations at the smoothing scale indicated by the x-axis. For clarity, the shot noise error bars of the signal are not plotted, their amplitude can be read in Fig. 8. From bottom to top, the dashed lines correspond to: model (1), model (2) and model (3) as given in Table 2. The shot-noise error bars of the signal are in fact comparable in amplitude to the cosmic variance error bars of model (2). We show also a cluster-normalized [FORMULA] model (dotted line) with [FORMULA], [FORMULA], and a CDM power spectrum with [FORMULA]. This model was not obtained from a simulation, but computed using the non-linear power spectrum using the Peacock & Dodds 1996 formula.

[FIGURE] Fig. 11. Star ellipticities of all the survey before (top panel) and after (bottom panel) the correction. After correction, the star ellipticity is randomly distributed around zero, as expected.


Table 2. List of the ray tracing simulations we used (see Jain et al. 1999 for details). The redshift of the sources is 1.

Our measurements are in agreement with the cluster normalized model (2). Also plotted is the theoretical prediction of a [FORMULA]CDM model, with [FORMULA], [FORMULA], [FORMULA] and a redshift of the sources [FORMULA]. It shows that the low-[FORMULA] model is also in good agreement with the data, which means that weak gravitational lensing provides cosmological constraints similar to the cluster abundance results (Eke et al. 1996, Blanchard et al. 1999): the second moment of the shear measures a combination of [FORMULA] and [FORMULA] (see Eq. 8). A measure of the third moment of the convergence would break the [FORMULA]-[FORMULA] degeneracy, but this requires more data (see Bernardeau et al. 1997, Van Waerbeke et al. 1999, Jain et al. 1999). It should also be noted that for the simulations, we have considered cold dark matter models with shape parameter [FORMULA]; higher values of [FORMULA] increase the theoretical predictions on scales of interest, e.g. the [FORMULA], [FORMULA] model would be ruled out even more strongly. We conclude that our analysis is consistent with the current favored cosmological models, although we cannot yet reject other models with high significance. Since we have only analyzed 2 square degrees of the survey, with forthcoming larger surveys we should be able to set strong constraints on the cosmological models as discussed below.

Due to the imprecise knowledge of the redshift distribution in our data, the interpretation might still be subject to modifications. The final state of our survey in 4 colors will however permit the measurement of this distribution by estimating photometric redshifts for the source galaxies.

[FIGURE] Fig. 12. Uncorrected (left) and corrected (right) star ellipticities for FIELD F14P1. The dashed cross shows the location of the optical center. Frames are graduated in pixels. The reference stick at the top-left of the frame shows the amplitude of a [FORMULA] distortion. This length of reference applies also for Figs. 13 to 19.

[FIGURE] Fig. 13. Same as Fig. 12 for FIELD F14P2.

[FIGURE] Fig. 14. Same as Fig. 12 for FIELD F14P3.

[FIGURE] Fig. 15. Same as Fig. 12 for FIELD 03hrie.

[FIGURE] Fig. 16. Same as Fig. 12 for FIELD SA57.

[FIGURE] Fig. 17. Same as Fig. 12 for FIELD a1942.

[FIGURE] Fig. 18. Same as Fig. 12 for FIELD F02P1.

[FIGURE] Fig. 19. Same as Fig. 12 for FIELD F02P4.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000