Cover picture: When a source, e. g., a QSO, lies behind a foreground galaxy, its light bundle is affected by the individual stars of this galaxy. This microlensingeffect, so far observed in at least one QSO (see Sect. 12.4), leads to a change in the flux we observe from the source, relative to an unlensed source. The flux magnification depends sensitively on the position of the source relative to the stars in the galaxy. Here we see the magnification as a function of the relative source position; red and yellow indicates high magnification, green and blue low magnification. The superimposed white curves are the caustics produced by the stars, projected into the source plane. The figure (which is taken from Wambsganss, Witt, and Schneider. Astr. Astophys., 258. 591 (1992)) has been produced by combining the ray-shooting method (Sects. 10.6. and 11.2.5) with the parametric representation of caustics (Sect. 8.3.4). The parameters for the star field are 1(.=0.5. y =1(,= O. with all stars having the same mass. The shape of the acoustics is analyzed in Chap. 6.

According to the theory of general relativity, masses deflect light in a way similar to convex glass lenses. This gravitational lensing effect is astigmatic, giving rise to image distortions. These distortions allow to quantify cosmic structures statistically on a broad range of scales, and to map the spatial distribution of dark and visible matter. We summarise the theory of weak gravitational lensing and review applications to galaxies, galaxy clusters and larger-scale structures in the Universe.

Cover picture: When a source, e.g., a QSO, lies behind a foreground galaxy, its light boundle is affected by the individual stars ofthis galaxy. This microlensing effect, so far observed in at least one QSO (see Sect. 12.4), leads to a change in the flux we observe from the source, relative to an lInlensed source. The flux magnification depends sensitively on the position of the source relative to the stars in the galaxy. Here we see the magnification as a function of the relative source position: red and yellow indicates high magnification, green and blue low magnification. The superimposed white curves are the caustics produced by the stars, projected into the source plane. The figure (taken from Wambsganss, Witt, and Schneider, Asfr. Astrophys., 258, 591 (1992)) has been produced by combining the ray-shooting method (Sects. 10.6 and 11.2.5) with the parametric representation of caustics (Sect. 8.3.4). The parameters for the star field are K. = 0.5, ' Y = Kc= 0, with all stars having the same mass. The shape of the acoustics is analyzed in Chap. 6.

Aims. The maximum-likelihood method is the standard approach to obtain model fits to observational data and the corresponding confidence regions. We investigate possible sources of bias in the log-likelihood function and its subsequent analysis, focusing on estimators of the inverse covariance matrix. Furthermore, we study under which circumstances the estimated covariance matrix is invertible. Methods. We perform Monte-Carlo simulations to investigate the behaviour of estimators for the inverse covariance matrix, depending on the number of independent data sets and the number of variables of the data vectors. Results. We find that the inverse of the maximum-likelihood estimator of the covariance is biased, the amount of bias depending on the ratio of the number of bins (data vector variables), p, to the number of data sets, n. This bias inevitably leads to an -in extreme cases catastrophic -underestimation of the size of confidence regions. We report on a method to remove this bias for the idealised case of Gaussian noise and statistically independent data vectors. Moreover, we demonstrate that marginalisation over parameters introduces a bias into the marginalised log-likelihood function. Measures of the sizes of confidence regions suffer from the same problem. Furthermore, we give an analytic proof for the fact that the estimated covariance matrix is singular if p > n.

We consider here a new statistical measure for cosmic shear, the aperture mass Map, 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 provides a well localized estimate of the power spectrum at wavenumbers s~5/theta. We calculate 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 relative to the linear growth on scales below ~0.5 degree. 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 the shear in the frame of 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 are constructed and their dispersions calculated. We show that one sq.deg. of a high-quality image is sufficient to detect the cosmic shear with the Map-statistics on scales below ~10 arcmin, and to estimate its amplitude with an accuracy of ~30% on scales below ~5 arcmin. (abbreviated)Comment: 34 pages, plain TeX, including 9 figures, submitted to MNRA