5 Results and comparison to cosmological models

Figure 3: Top-hat smoothed variance of the shear (points with error bars). The three models correspond to $(\Omega _0,\Lambda ,\sigma _8)=(0.3,0,0.9),(0.3,0.7,0.9),(1,0,0.6)$ for the short-dashed, solid and long-dashed lines respectively. The power spectrum is a CDM-model with $\Gamma =0.21$. The error bars correspond to the dispersion of the variance measured from 200 realizations of the data set with randomized orientations of the galaxy ellipticities.

Figure 4: The aperture mass statistic for the same models as in Fig. 3. The lower panel plots the R-mode, obtained by making a 45 degree rotation as described in the text. There is no significant detection for $\theta > 5~{\rm arcmin}$ (corresponding to an effective angular scale of 1', as discussed in the text), which shows the low level of contamination by galaxy intrinsic alignment and/or residual systematics.

Figure 5: Shear correlation function $\langle\gamma\gamma\rangle_\theta$. The models are the same as in Fig. 3. The lower panel uses a log-scale for the x-axis to highlight the small scale details.

Figure 6: Top panel: tangential shear correlation function $\langle\gamma_{\rm t}\gamma_{\rm t}\rangle_\theta$. Bottom panel: radial shear correlation function $\langle\gamma_{\rm r}\gamma_{\rm r}\rangle_\theta$. The models are the same as in Fig. 3.

In this section we present our measurements of the 2-point correlations of the shear using the different estimators defined above. Figures 3 to 8 show the results for the different estimators: the variance in Fig. 3, the mass aperture statistic in Fig. 4, the shear correlation function in Fig. 5, the radial and tangential shear correlations in Fig. 6, and the cross-correlation of the radial and tangential shear in Fig. 8. Along with the measurements we show the predictions of three cosmological models which are representative of an open model, a flat model with cosmological constant, and an Einstein-de Sitter model. The amplitude of mass fluctuations in these models is normalized to the abundance of galaxy clusters. The three models are char- acterized by the values of $\Omega_0, \Lambda$ and $\sigma_8$ as follows:

• short-dashed line: $\Omega _0=0.3$, $\Lambda =0$, $\sigma_8=0.9$
• solid line: $\Omega _0=0.3$, $\Lambda=0.7$, $\sigma_8=0.9$
• long-dashed line: $\Omega_0=1$, $\Lambda =0$, $\sigma_8=0.6$.

The power spectrum is taken to be a cold dark matter (CDM) power spectrum with shape parameter $\Gamma =0.21$. The predictions for shear correlations are computed using the non-linear evolution of the power spectrum using the Peacock & Dodds (1996) fitting formula. It is assumed that the source redshift distribution follows Eq. (2) with $(z_0,\alpha ,\beta )=(0.8,2,1.5)$, which corresponds to a mean redshift of 1.2.

It is reassuring that the different statistics agree with each other in their comparison with the model predictions. These statistics weight the data in different ways and are susceptible to different kinds of systematic errors. The consistency of all the 2-point estimators suggests that the level of systematics in the data is low compared to the signal. A further test for systematics is provided by measuring the cross-correlation function $\langle\gamma_{\rm t}\gamma_{\rm r}\rangle_\theta$, which should be zero for a signal due to gravitational lensing. It is shown in Fig. 8 that it is indeed consistent with zero at all scales. The figure also shows the cross-correlation obtained when the anisotropic contamination of the PSF is not corrected - clearly such a correction is crucial in measuring the lensing signal.

The lower panel of Fig. 4 shows the R-mode of the mass aperture statistic. As this statistic uses a compensated filter, the scale beyond which the measured R-mode is consistent with zero (5' on the plot) corresponds to an effective angular scale $\theta \simeq 1'$. This places an upper limit on measured shear correlations due to the intrinsic alignment of galaxies, given the redshift distribution of the sources. The vanishing of $\langle M_\perp^2\rangle$ for effective angular scales larger than 1' strongly supports our conclusion that the level of residual systematics is low: this is a very hard test to pass, as it means that the signal is produced by a pure scalar field, which need not be the case for systematics. We checked that $M_\perp^2$is Gaussian distributed with a zero average all over the survey, which is what we would expect from a pure noise realisation. For scales below 5' on the plot, the R-mode is not consistent with zero at the 2-$\sigma$ level. Since the cross-correlation $\langle\gamma_{\rm t}\gamma_{\rm r}\rangle_\theta$ is consistent with zero at this scale, the source of the R-mode is probably not a residual systematic caused by an imperfect PSF correction. Rather, it might be due to the effect of intrinsic alignments (Crittenden et al. 2000b).

The error bars shown in Figs. 3 to 8 are calculated from a measurement of the different statistics in 200 realizations of the data set, with randomized orientations of the galaxies. We measured the sample variance from ray-tracing simulations (Jain et al. 2000) and find that it is smaller than $20\%$ of the noise error bars shown here (see Van Waerbeke et al. 1999 where the sample variance has been calculated for surveys with similar geometry), therefore we have not included it in our figures. Figure 7 shows an estimate of the sample variance for the rms shear using a compact 6.5 deg²ray-tracing simulation (Jain et al. 2000). This figure shows that the sample variance is about one order of magnitude smaller than $\langle \kappa^2\rangle^{1/2}$for the range of scales of interest. Hence our errors are not dominated by sample variance, as was the case in the first detections of cosmic shear. As the probed angular scales approach the size of the fields (which is ${\sim} 30'$with the CFH12K camera) the sample variance becomes larger. This could be responsible for the small fluctuations in the measured correlations in Figs. 5 and 6 for scales larger than 24'.

Figure 7: Shear rms $\langle \gamma^2\rangle^{1/2}$(solid line) measured in a ray-tracing simulation (Jain et al. 2000) for the open $\Omega _0=0.3$ model. The dashed line is the sample variance of the shear rms measured from 7 different realisations of the mass distribution for a survey of 6.5 deg2.

Figure 8: Shear cross-correlation function $\langle\gamma_{\rm t}\gamma_{\rm r}\rangle_\theta$. The signal should vanish if the data are not contaminated by systematics. As a comparison, the open circles show the same cross-correlation function computed from the galaxy ellipticities where the anisotropic correction of the PSF has been skipped.