A Method for Weak Lensing Observations

We develop and test a method for measuring the gravitational lensing induced distortion of faint background galaxies. We first describe how we locate the galaxies and measure a 2-component `polarisation' or ellipticity statistic $e_\alpha$ whose expectation value should be proportional to the gravitational shear $\gamma_\alpha$. We then show that an anisotropic instrumental psf perturbs the polarisation by $\delta e_\alpha = P^s_{\alpha\beta} p_\beta$, where $p_\alpha$ is a measure of the psf anisotropy and $P^s_{\alpha\beta}$ is the `linearised smear polarisability tensor'. By estimating $P^s_{\alpha\beta}$ for each object we can determine $p_\alpha$ from the foreground stars and apply a correction $-P^s_{\alpha\beta}p_\beta$ to the galaxies. We test this procedure using deep high-resolution images from HST which are smeared with an anisotropic psf and then have noise added to simulate ground-based observations. We find that the procedure works very well. A similar analysis yields a linear shear polarisability tensor $P^\gamma_{\alpha\beta}$ which describes the response to a gravitational shear. This calibrates the polarisation-shear relation, but only for galaxies which are well resolved. To empirically calibrate the effect of seeing on the smaller galaxies we artificially stretch HST images to simulate lensing and then degrade them as before. These experiments provide a rigorous and exacting test of the method under realistic conditions. They show that it is possible to remove the effect of instrumental psf anisotropy, and that the method provides an efficient and quantitative measurement of the gravitational shear.