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 distri bution in the Universe, offers the opportunity to measure the power sp ectrum of the cosmic density fluctuations without any reference to the relation of dark matter to luminous tracers. We consider here a new s tatistical measure for cosmic shear, the aperture mass M-ap(theta) whi ch is defined as a spatially filtered projected density field and whic h can be measured directly from the image distortions of high-redshift galaxies. By selecting an appropriate spatial filter function, the di spersion of the aperture mass is a convolution of the power spectrum o f the projected density field with a narrow kernel, so that [M-ap(2)(t heta)] provides a well-localized estimate of the power spectrum at wav enumbers s similar to 5/theta. We calculate [M-ap(2)] for various cosm ological models, using the fully non-linear power spectrum of the cosm ic density fluctuations. The non-linear evolution yields a significant increase of [M-ap(2)] relative to the linear growth on scales below s imilar to 0.degrees 5. The third-order moment of M-ap can be used to d efine a skewness, which is a measure of the non-Gaussianity of the den sity 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 sp ectrum. Several practical estimates for [M-ap(2)] are constructed and their dispersions calculated. On scales below a few arcminutes, the in trinsic 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 adjac ent apertures are virtually uncorrelated, which implies that an image with side-length L can yield [L/(2 theta)](2) mutually independent est imates for M-ap. We show that one square degree of a high-quality imag e is sufficient to detect the cosmic shear with the M,p-statistic on s cales below similar to 10 arcmin, and to estimate its amplitude with a n accuracy of similar to 30 per cent on scales below similar to 5 arcm in.