Weak gravitational lensing of distant galaxies

We analyze the two-point statistics of the gravitationally induced ellipticities of distant galaxy images which provide a direct probe of the mass fluctuation spectrum P(k). The analysis extends previous work in the following respects: (1) We explore the relation between the ellipticity pattern of the background galaxies and the projected density of foreground matter. This yields both useful statistical correlations and also an approximate method for directly imaging foreground mass concentrations. (2) We calculate angular power spectra, using the analogue of Limber's equation in the Fourier domain, as well as angular correlation functions. We find that there are some characteristic features of the ellipticity correlations that appear most clearly in the Fourier representation. (3) The analysis is valid for arbitrary evolution of P(k) and for an arbitrary distance distribution for the galaxies. We show that the size of the effect on very large scales is tightly constrained by microwave anisotropy limits, but that the density fluctuations inferred from observed peculiar motions suggest interestingly large effects at angular scales below about a degree. We compute angular power spectra and correlation functions for a number of illustrative models for P(k). We discuss methods for estimating the various statistics, estimate their statistical uncertainty, and discuss the feasibility of this test.