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 arising from 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 arising from 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 held are a direct measure of the power spectrum of density fluctuations.