Analysis of two-point statistics of cosmic shear

A&A 396, 1-19 (2002)
DOI: 10.1051/0004-6361:20021341

I. Estimators and covariances

P. Schneider^{1, 2}, L. van Waerbeke^{3, 4}, M. Kilbinger¹ and Y. Mellier^{3, 5}

¹ Institut f. Astrophysik u. Extr. Forschung, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
² Max-Planck-Institut f. Astrophysik, Postfach 1317, 85741 Garching, Germany
³ Institute d'Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France
⁴ Canadian Institute for Theoretical Astrophysics, 60 St Georges Str., Toronto, M5S 3H8 Ontario, Canada
⁵ Observatoire de Paris, DEMIRM/LERMA, 61 avenue de l'Observatoire, 75014 Paris, France

(Received 13 June 2002 / Accepted 12 September 2002)

Abstract
Recently, cosmic shear, the weak lensing effect by the inhomogeneous matter distribution in the Universe, has not only been detected by several groups, but the observational results have been used to derive constraints on cosmological parameters. For this purpose, several cosmic shear statistics have been employed. As shown recently, all second-order statistical measures can be expressed in terms of the two-point correlation functions of the shear, which thus represents the basic quantity; also, from a practical point-of-view, the two-point correlation functions are easiest to obtain from observational data which typically have complicated geometry. We derive in this paper expressions for the covariance matrix of the cosmic shear two-point correlation functions which are readily applied to any survey geometry. Furthermore, we consider the more special case of a simple survey geometry which allows us to obtain approximations for the covariance matrix in terms of integrals which are readily evaluated numerically. These results are then used to study the covariance of the aperture mass dispersion which has been employed earlier in quantitative cosmic shear analyses. We show that the aperture mass dispersion, measured at two different angular scales, quickly decorrelates with the ratio of the scales. Inverting the relation between the shear two-point correlation functions and the power spectrum of the underlying projected matter distribution, we construct estimators for the power spectrum and for the band powers, and show that they yields accurate approximations; in particular, the correlation between band powers at different wave numbers is quite weak. The covariance matrix of the shear correlation function is then used to investigate the expected accuracy of cosmological parameter estimates from cosmic shear surveys. Depending on the use of prior information, e.g. from CMB measurements, cosmic shear can yield very accurate determinations of several cosmological parameters, in particular the normalization $\sigma_8$ of the power spectrum of the matter distribution, the matter density parameter $\Omega_{\rm m}$ , and the shape parameter $\Gamma$ .

Key words: dark matter -- gravitational lensing -- large-scale structure of the Universe

Offprint request: P. Schneider, peter@astro.uni-bonn.de