8 Conclusion

Using 6.5 deg² of the VIRMOS-Descart survey in progress at the CFHT, we were able to measure various 2-point correlation statistics of cosmic shear. The top-hat variance, the aperture mass statistic and different shear correlation functions gave consistent results over a wide range of scales. Further tests of the lensing origin of the signal exploiting the scalar nature of the gravitational potential were also convincingly verified. We demonstrated that the level of systematics, in particular the intrinsic alignment of galaxies, is likely to be small, and does not contribute to the signal for scales larger than 1'. For scales smaller than 1' we have either ignored the measurements, or increased the error bars by the amount of signal found in the B-mode analysis which will be presented elsewhere. We believe that these results demonstrate the significance of our detection of shear correlations due to gravitational lensing. The quality of the data and the adequate size of our survey allow us to put some constrains on the cosmological models. We obtained tight constraints on the cosmological parameters $\Omega_0$ and $\sigma_8$ modulo, the uncertainties in the value of the power spectrum slope $\Gamma$ and in the source redshift distribution. We found that different values of $\Gamma$ significantly affects the degeneracy breaking in the $\Omega_0$, $\sigma_8$ plane. Only the use of external priors can allow robust and independent constraints on these parameters. A detailed analysis based on a marginalisation of $\Gamma$ and the observed redshift distribution (using photometric redshifts) will be done in a forthcoming paper. Nevertheless these results suggest that high precision measurements can be made with larger lensing surveys on a much larger set of cosmological parameters. Clearly, weak lensing surveys will play an important role in measuring the cosmological parameters together with Cosmic Microwave Background, Supernovae, and Large Scale Structures (Hu & Tegmark 1999).

The final stage of the VIRMOS survey is to accomplish 16 deg² in patches of 4 deg², 4 colors each, thus allowing the possibility to use the photometric redshifts of the galaxies. The use of photometric (or spectroscopic) redshifts will be useful to put robust constraints on the cosmological parameters and improve the scientific interpretation of cosmic shear (e.g. doing tomography as in Hu 1999) but also to measure the intrinsic alignments itself (which can be used to constrain galaxy formation models for instance).

The constraints obtained so far are within a framework of structure formation through gravitational instability with Gaussian initial conditions and Cold Dark Matter. As the amount of observations increases and the measurement quality improves, the first hints of the shape of the power spectrum will be soon available. It opens new means of really testing the formation mechanisms of the large-scale structure and the cosmological parameters beyond the standard model (Uzan & Bernardeau 2000).

Over the last two years, we have seen the transition from the detection of the weak lensing signal to the first measurements of cosmological parameters from it. The agreement between theoretical predictions and observational results with such a high precision indicates that the measurement of cosmic shear statistics is becoming a mature cosmological tool. Many surveys are under way or scheduled for the next 5 years. They will use larger panoramic cameras than the CFH12K, and will cover solid angles 10 to 100 times wider than this work. The results of this work give us confidence that cosmic shear statistics will provide valuable measurements of cosmological parameters, probe the biasing of mass/light, produce maps of the dark matter distribution and reconstruct its power spectrum.

Acknowledgements

We are grateful to S. Colombi, U.-L. Pen, D. Pogosyan, S. Prunet, I. Szapudi and S. White for useful discussions related to statistics. We thank H. Hoekstra for sharing his results prior to publication. This work was supported by the TMR Network "Gravitational Lensing: New Constraints on Cosmology and the Distribution of Dark Matter'' of the EC under contract No. ERBFMRX-CT97-0172, and a PROCOPE grant No. 9723878 by the DAAD and the A.P.A.P.E. We thank the TERAPIX data center for providing its facilities for the data reduction of CFH12K images.