Theory and Statistics of Weak Lensing from Large-Scale Mass Inhomogeneities

Weak lensing by large-scale mass inhomogeneities in the Universe induces correlations in the observed ellipticities of distant sources. We first review the harmonic analysis and statistics required of these correlations and discuss calculations for the predicted signal. We consider the ellipticity correlation function, the mean-square ellipticity, the ellipticity power spectrum, and a global maximum-likelihood analysis to isolate a weak-lensing signal from the data. Estimates for the sensitivity of a survey of a given area, surface density, and mean intrinsic source ellipticity are presented. We then apply our results to the FIRST radio-source survey. We predict an rms ellipticity of roughly 0.011 in $1^\circ \times 1^\circ$ pixels and 0.018 in $20' \times 20'$ pixels if the power spectrum is normalized to $\sigma_8 \Omega^{0.53} = 0.6$, as indicated by the cluster abundance. The signal is significantly larger in some models if the power spectrum is normalized instead to the COBE anisotropy. The uncertainty in the predictions from imprecise knowledge of the FIRST redshift distribution is about 25% in the rms ellipticity. We show that FIRST should be able to make a statistically significant detection of a weak-lensing signal for cluster-abundance-normalized power spectra.