A new measure for cosmic shear

Cosmic shear, i.e. the distortion of images of high-redshift galaxies through the tidal gravitational field of the large-scale matter distribution in the Universe, offers the opportunity to measure the power spectrum of the cosmic density fluctuations without any reference to the relation of dark matter to luminous tracers. We consider here a new statistical measure for cosmic shear, the aperture mass M-ap(theta) which is defined as a spatially filtered projected density field and which can be measured directly from the image distortions of high-redshift galaxies. By selecting an appropriate spatial filter function, the dispersion of the aperture mass is a convolution of the power spectrum of the projected density field with a narrow kernel, so that [M-ap(2)(theta)] provides a well-localized estimate of the power spectrum at wavenumbers s similar to 5/theta. We calculate [M-ap(2)] for various cosmological models, using the fully non-linear power spectrum of the cosmic density fluctuations. The non-linear evolution yields a significant increase of [M-ap(2)] relative to the linear growth on scales below similar to 0.degrees 5. The third-order moment of M-ap can be used to define a skewness, which is a measure of the non-Gaussianity of the density held. We present the first calculation of the skewness of cosmic shear in the framework of the quasi-linear theory of structure growth. We show that it yields a sensitive measure of the cosmological model; in particular, it is independent of the normalization of the power spectrum. Several practical estimates for [M-ap(2)] are constructed and their dispersions calculated. On scales below a few arcminutes, the intrinsic ellipticity distribution of galaxies is the dominant source of noise, whereas on larger scales the cosmic variance becomes the most important contribution. We show that measurements of M-ap in two adjacent apertures are virtually uncorrelated, which implies that an image with side-length L can yield [L/(2 theta)](2) mutually independent estimates for M-ap. We show that one square degree of a high-quality image is sufficient to detect the cosmic shear with the M,p-statistic on scales below similar to 10 arcmin, and to estimate its amplitude with an accuracy of similar to 30 per cent on scales below similar to 5 arcmin.