Cosmic shear is considered one of the most powerful methods for studying the properties of Dark Energy in the Universe. As a standard method, the two-point correlation functions $xi_pm(theta)$ of the cosmic shear field are used as statistical measure
s for the shear field. In order to separate the observed shear into E- and B-modes, the latter being most likely produced by remaining systematics in the data set and/or intrinsic alignment effects, several statistics have been defined before. Here we aim at a complete E-/B-mode decomposition of the cosmic shear information contained in the $xi_pm$ on a finite angular interval. We construct two sets of such E-/B-mode measures, namely Complete Orthogonal Sets of E-/B-mode Integrals (COSEBIs), characterized by weight functions between the $xi_pm$ and the COSEBIs which are polynomials in $theta$ or polynomials in $ln(theta)$, respectively. Considering the likelihood in cosmological parameter space, constructed from the COSEBIs, we study their information contents. We show that the information grows with the number of COSEBI modes taken into account, and that an asymptotic limit is reached which defines the maximum available information in the E-mode component of the $xi_pm$. We show that this limit is reached the earlier (i.e., for a smaller number of modes considered) the narrower the angular range is over which $xi_pm$ are measured, and it is reached much earlier for logarithmic weight functions. For example, for $xi_pm$ on the interval $1le thetale 400$, the asymptotic limit for the parameter pair $(Omega_m, sigma_8)$ is reached for $sim 25$ modes in the linear case, but already for 5 modes in the logarithmic case. The COSEBIs form a natural discrete set of quantities, which we suggest as method of choice in future cosmic shear likelihood analyses.
We introduce an optimized data vector of cosmic shear measures (N). This data vector has high information content, is not sensitive against B-mode contamination and only shows small correlation between data points of different angular scales. We show
that a data vector of the two-point correlation function (2PCF) in general contains more information on cosmological parameters compared to a data vector of the aperture mass dispersion. Reason for this is the fact that <M_ap^2> lacks the information of the convergence power spectrum (P_kappa) on large angular scales, which is contained in the 2PCF data vector. Therefore we create a combined data vector N, which retains the advantages of <M_ap^2> and in addition is also sensitive to the large-scale information of P_kappa. We compare the information content of the three data vectors by performing a detailed likelihood analysis and use ray-tracing simulations to derive the covariance matrices. In the last part of the paper we contaminate all data vectors with B-modes on small angular scales and examine their robustness against this contamination.The combined data vector strongly improves constraints on cosmological parameters compared to <M_ap^2>. Although, in case of a pure E-mode signal the information content of the 2PCF is higher, in the more realistic case where B-modes are present the 2PCF data vector is strongly contaminated and yields biased cosmological parameter estimates. N shows to be robust against this contamination. Furthermore the individual data points of N show a much smaller correlation compared to the 2PCF leading to an almost diagonal covariance matrix.
