A new measure for cosmic shear

Abstract

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 Map(θ), 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 〈M2ap(θ)〉 provides a well-localized estimate of the power spectrum at wavenumbers s ∼ 5/θ. We calculate 〈M2ap〉 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 〈M2ap〉 relative to the linear growth on scales below ∼ 0°.5. The third-order moment of Map can be used to define a skewness, which is a measure of the non-Gaussianity of the density field. 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 〈M2ap〉 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 Map in two adjacent apertures are virtually uncorrelated, which implies that an image with side-length L can yield [L/(2θ)]2 mutually independent estimates for Map. We show that one square degree of a high-quality image is sufficient to detect the cosmic shear with the Map-statistic on scales below ∼ 10 arcmin, and to estimate its amplitude with an accuracy of ∼ 30 per cent on scales below ∼ 5 arcmin.