Weak Lensing and Cosmology

We explore the dependence of weak lensing phenomena on the backgroundcosmology. We first generalise the relation between $P_\psi(\omega)$, theangular power spectrum of the distortion, and the power spectrum of densityfluctuations to non-flat cosmologies. We then compute $P_\psi$ for variousillustrative models. A useful cosmological discriminator is the growth of$P_\psi$ with source redshift which is much stronger in low matter densitymodels, and especially in $\Lambda$-dominated models. With even crude redshiftinformation (say from broad band colours) it should be possible to constrainthe cosmological world model. The amplitude of $P_\psi(\omega)$ is also quitesensitive to the cosmology, but requires a reliable external normalisation forthe mass fluctuations. If one normalises to galaxy clustering, with $M/L$ fixedby small-scale galaxy dynamics, then low density models predict a much strongerdistortion. If, however, one normalises to large-scale bulk-flows, thepredicted distortion for sources at redshifts $Z_s \sim 1-3$ is ratherinsensitive to the background cosmology. The signals predicted here can bedetected at a very high level of significance with a photometric surveycovering say 10 square degrees, but sparse sampling is needed to avoid largesampling variance and we discuss the factors influencing the design of anoptimum survey. Turning to weak lensing by clusters we find that for high lensredshifts ($Z_l\simeq1$) the critical density is substantially reduced in$\Lambda$ models, but that the ratio of the shear or convergence to thevelocity dispersions or X-ray temperature of clusters is only very weaklydependent on the cosmology.