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Weak lensing by large scale structure induces correlated ellipticities in theimages of distant galaxies. The two-point correlation is determined by thematter power spectrum along the line of sight. We use the fully nonlinearevolution of the power spectrum to compute the predicted ellipticitycorrelation. We present results for different measures of the second moment forangular scales \theta \simeq 1'-3 degrees and for alternative normalizations ofthe power spectrum, in order to explore the best strategy for constraining thecosmological parameters. Normalizing to observed cluster abundance the rmsamplitude of ellipticity within a 15' radius is \simeq 0.01 z_s^{0.6}, almostindependent of the cosmological model, with z_s being the median redshift ofbackground galaxies.  Nonlinear effects in the evolution of the power spectrum significantlyenhance the ellipticity for \theta < 10' -- on 1' the rms ellipticity is \simeq0.05, which is nearly twice the linear prediction. This enhancement means thatthe signal to noise for the ellipticity is only weakly increasing with anglefor 2'< \theta < 2 degrees, unlike the expectation from linear theory that itis strongly peaked on degree scales. The scaling with cosmological parametersalso changes due to nonlinear effects. By measuring the correlations on small(nonlinear) and large (linear) angular scales, different cosmologicalparameters can be independently constrained to obtain a model independentestimate of both power spectrum amplitude and matter density \Omega_m.Nonlinear effects also modify the probability distribution of the ellipticity.Using second order perturbation theory we find that over most of the range ofinterest there are significant deviations from a normal distribution.Comment: 38 pages, 11 figures included. Extended discussion of observational  prospects, matches accepted version to appear in Ap

Cosmological Model Predictions for Weak Lensing: Linear and Nonlinear Regimes