Weak Lensing by Large Scale Structure in Open, Flat, and Closed Universes

Weak lensing is the distortion (polarization) of images of distant objects, such as high redshift galaxies, by gravitational fields in the limit where the distortion is small. Gravitational potential fluctuations due to large scale structure cause correlated distortions of the images of high redshift galaxies. These distortions are observable with current large telescopes and instrumentation. In a Friedmann-Robertson-Walker (FRW) metric I calculate the weak lensing pattern due to large scale structure for arbitrary $\Omega_0$ and zero cosmological constant $\Lambda$. For a given cosmological model, specified by $\Omega_0$ and a power spectrum of density fluctuations, I calculate the statistical properties of the polarization field for an arbitrary redshift source distribution in a simple closed form. It is shown that for low redshift $z$ of the sources, the polarization amplitude is proportional to $\Omega_0$, while at higher redshift the polarization measures the value of $\Omega(z)$, where $z$ is the characteristic source redshift. Moreover, the statistics of the polarization field are a direct measure of the power spectrum of density fluctuations.