Weak Lensing and Cosmology

We explore the dependence of weak lensing phenomena on the background cosmology. We first generalise the relation between $P_\psi(\omega)$, the angular power spectrum of the distortion, and the power spectrum of density fluctuations to non-flat 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 broad band colours) 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 normalisation for the mass fluctuations. If one normalises to galaxy clustering, with $M/L$ fixed by small-scale galaxy dynamics, then low density models predict a much stronger distortion. If, however, one normalises to large-scale bulk-flows, the predicted distortion for sources at redshifts $Z_s \sim 1-3$ 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 square degrees, but sparse sampling is needed to avoid large sampling variance and we discuss the factors influencing the design of an optimum survey. Turning to weak lensing by clusters we find that for high lens redshifts ($Z_l\simeq1$) 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.