COSEBIs: Extracting the full E-/B-mode information from cosmic shear
  correlation functions
  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 measures 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 $1'\le \theta\le
400'$, 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.
