Cosmic shears should not be measured in conventional ways

A long-standing problem in weak lensing is about how to construct cosmic shear estimators from galaxy images. Conventional methods average over a single quantity per galaxy to estimate each shear component. We show that any such shear estimators must reduce to a highly non-linear form when the galaxy image is described by three parameters (pure ellipse), even in the absence of the point spread function (PSF). In the presence of the PSF, we argue that this class of shear estimators are not likely to exist. Alternatively, we propose a new way of measuring the cosmic shear: instead of averaging over a single value from each galaxy, we average over two numbers, and then take the ratio to estimate the shear component. In particular, the two numbers correspond to the numerator and denominator that generate the quadrupole moments of the galaxy image in Fourier space, as proposed by Zhang. This yields a statistically unbiased estimate of the shear component. Consequently, measurements of the n-point spatial correlations of the shear fields should also be modified: one needs to derive the ratio of the two correlation functions to get the desired unbiased shear correlation.