Weak lensing statistics as a probe of Ω and power spectrum

The possibility of detecting weak lensing effects from deep wide field imaging surveys has opened new means of probing the large-scale structure of the Universe and measuring cosmological parameters. In this paper we present a systematic study of the expected dependence of the low order moments of the filtered gravitational local convergence on the power spectrum of the density fluctuations and on the cosmological parameters Ω₀ and A. The results show a significant dependence on all these parameters. Though we note that this degeneracy could be partially raised by considering two populations of sources, at different redshifts, computing the third moment is more promising since it is expected, in the quasi-linear regime and for Gaussian initial conditions, to be only Ω₀ dependent (with a slight degeneracy with Λ) when it is correctly expressed in terms of the second moment. More precisely we show that the variance of the convergence vanes approximately as P(k) Ω^1.5 ₀ z^1.5 _s, whereas the skewness varies as Ω^0.8 ₀ z^-1.35 _s, where P(k) is the projected power spectrum and z_s the redshift of the sources. Thus, used jointly they can provide both P(k) and Ω₀ However, the dependence on the redshift of the sources is large and could be a major concern for a practical implementation. We have estimated the errors expected for these parameters in a realistic scenario and sketched what would be the observational requirements for doing such measurements. A more detailed study of an observational strategy is left for a second paper.