Weak lensing and cosmology

We explore the dependence of weak lensing phenomena on the background cosmology. We first generalize the relation between P-psi(omega), the angular power spectrum of the distortion, and the power spectrum of density fluctuations to nonflat cosmologies. We then compute P-psi for various illustrative models. A useful cosmological discriminator is the growth of P-psi with source redshift, which is much stronger in low-matter density models and especially in Lambda-dominated models. With even crude redshift information (say from broadband colors), it should be possible to constrain the cosmological world model. The amplitude of P-psi(omega) is also quite sensitive to the cosmology, but requires a reliable external normalization for the mass fluctuations. If one normalizes to galaxy clustering, with M/L fixed by small-scale galaxy dynamics, then low-density models predict a much stronger distortion. If, however, one normalizes to large-scale bulk flows, the predicted distortion for sources at redshifts Z(s) similar to 1-3 is rather insensitive to the background cosmology. The signals predicted here can be detected at a very high level of significance with a photometric survey covering, say, 10 deg(2), but sparse sampling is needed to avoid large sampling variance. We discuss the factors influencing the design of an optimum survey. Turning to weak lensing by clusters, we find that for high lens redshifts (Z(l) similar or equal to 1) the critical density is substantially reduced in Lambda-models, but that the ratio of the shear or convergence to the velocity dispersions or X-ray temperature of clusters is only very weakly dependent on the cosmology.