COSEBIs: Extracting the full E-/B-mode information from cosmic shear correlation functions

TL;DR

This work introduces COSEBIs (Complete Orthogonal Sets of E-/B-mode Integrals) to achieve a complete E-/B-mode decomposition of cosmic shear information from the two-point correlation functions on a finite angular interval. By constructing two families of weight functions—polynomial in θ and in ln θ—COSEBIs provide a natural discrete, orthogonal basis {E_n} (and {B_n}) whose information content grows with the number of modes and quickly saturates, especially for logarithmic weights. The authors derive the analytic form of polynomial-weight COSEBIs, develop a numerical approach for logarithmic weights, and show that the COSEBIs recover nearly all the information contained in the full 2PCFs while simplifying covariance estimation and mitigating extrapolation biases. They demonstrate, with a CFHTLS-like survey, that a small number of logarithmic COSEBI modes suffices to reach the asymptotic information limit, highlighting COSEBIs as a practical default for future cosmic shear likelihood analyses and enabling robust E-/B-mode separation even in the presence of B-mode systematics and baryonic uncertainties.

Abstract

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 $1'\le θ\le 400'$, 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.

Figures (12)

• Figure 1: The linear filter functions $T_{\pm n}(\vartheta)$ for $\vartheta_{\rm min}=1'$, $\vartheta_\mathrm{max}=400'$. Note that the shape of the curves depends only on the ratio ${\vartheta_{\rm min}}/{\vartheta_{\rm max}}$
• Figure 2: The functions $W_n$ as defined in Eq.(\ref{['eq:WnFilter']}) which relate the COSEBIs to the underlying power spectrum, calculated from the $T_{\pm n}$. The upper panel corresponds to $\vartheta_\mathrm{max}=400'$, whereas the lower panel is calculated using $\vartheta_\mathrm{max}=20'$, both for $\vartheta_{\rm min}=1'$
• Figure 3: Mathematica WolfMathe program to calculate the roots in Eq.(\ref{['eq:tpluslogprod']}) -- they are stored with 8 significant digits in the lower left halve of the table ROOTS. Furthermore, the array norm[n] contains the normalization coefficients $N_n$
• Figure 4: The logarithmic filter functions $T_{+n}^\mathrm{log}$ for $\vartheta_{\rm min}=1'$ and $\vartheta_\mathrm{max}=400'$. The left panel shows the function over the whole interval, whereas the right panel provides a more detailed view for small $\vartheta$
• Figure 5: The logarithmic filter functions $T_{-n}^\mathrm{log}$ for $\vartheta_{\rm min}=1'$ and $\vartheta_\mathrm{max}=400'$. As in Fig.\ref{['fig:T_log_plus']}, the left panel shows the function over the whole interval, whereas the right panel provides a more detailed view for small $\vartheta$