Analysis of two-point statistics of cosmic shear

Abstract

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 twopoint 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 σ 8 of the power spectrum of the matter distribution, the matter density parameter Ω m , and the shape parameter Γ.

Dependence of cosmic shear covariances on cosmology

“…For more details of this topic, the reader is referred to Bartelmann & Schneider (2001), Schneider et al (2002a), Schneider et al (2002b), Schneider (2004), andJoachimi et al (2008).…”

“…Only three of them are independent because C +− (ϑ i , ϑ j ) = C −+ (ϑ j , ϑ i ). Assuming a Gaussian shear field, the covariance of the 2PCF can be calculated analytically (Schneider et al 2002a;Joachimi et al 2008). There, the covariance is decomposed into three terms, namely the cosmic variance term (V), the pure shot noise term (S ), and the mixed term (M)…”

“…The corresponding expressions for V and M using the 2PCF are derived in Schneider et al (2002a). In this paper we only need the expressions for the mixed term, which read…”

“…Several methods are suggested in the literature and have been applied to cosmic shear data. An analytic expression for covariances assuming a Gaussian shear field was derived in Schneider et al (2002a) and confirmed in Joachimi et al (2008), who used a power spectrum approach that significantly reduces the computational effort in the calculation. This analytic expression has been used for parameter estimation in many surveys (e.g., van Waerbeke et al 2005, Semboloni et al 2006Hoekstra et al 2006).…”

Retrieving the three-dimensional matter power spectrum and galaxy biasing parameters from lensing tomography

“…This includes the source shape noise, lens position shot-noise, and cosmic variance of the actual signal. For an extensive discussion of the noise terms in ξ ± we refer to for example Schneider et al (2002). Towards this goal, we make N real = 4800 realisations of mock shear and lens catalogues in 2 × 2 deg 2 fields based on the previous survey parameters and the matter/galaxy clustering in our fiducial cosmological model; shear fields are Gaussian random fields and galaxy number density fields obey log-normal statistics.…”