Revisiting CFHTLenS cosmic shear: Optimal E/B mode decomposition using COSEBIs and compressed COSEBIs

TL;DR

This work re-analyzes the CFHTLenS weak-lensing data using COSEBIs to achieve complete E/B-mode separation and robust tests for systematics. It introduces compressed COSEBIs (CCOSEBIs) to manage covariance in tomographic analyses and presents the first tomographic COSEBIs measurements for CFHTLenS. The study finds significant B-modes on large scales in non-tomographic analyses, but B-modes become consistent with zero when using tomographic COSEBIs, illustrating the benefits of tomographic compression for preserving cosmological information while reducing data volume. The results highlight residual systematics in CFHTLenS on large scales, with implications for future surveys and the use of COSEBIs/CCOSEBIs in rigorous systematic tests.

Abstract

We present a re-analysis of the CFHTLenS weak gravitational lensing survey using Complete Orthogonal Sets of E/B-mode Integrals, known as COSEBIs. COSEBIs provide a complete set of functions to efficiently separate E-modes from B-modes and hence allow for robust and stringent tests for systematic errors in the data. This analysis reveals significant B-modes on large angular scales that were not previously seen using the standard E/B decomposition analyses. We find that the significance of the B-modes is enhanced when the data is split by galaxy type and analysed in tomographic redshift bins. Adding tomographic bins to the analysis increases the number of COSEBIs modes, which results in a less accurate estimation of the covariance matrix from a set of simulations. We therefore also present the first compressed COSEBIs analysis of survey data, where the COSEBIs modes are optimally combined based on their sensitivity to cosmological parameters. In this tomographic CCOSEBIs analysis we find the B-modes to be consistent with zero when the full range of angular scales are considered.

Figures (10)

• Figure 1: Measured COSEBIs from the CFHTLenS data for a single redshift bin using all galaxies. Three angular ranges are considered here. The dashed line shows the zero B-mode value. The $B_n$ modes (red circles) are shifted to the right for visual assistance. The $E_n$ (black squares) are compared with their theoretical values given the Planck (red dotted curve) and CFHTLenS+WMAP7 (blue solid curve) cosmologies. The CFHTLenS+WMAP7 theoretical values are best fit values for the $[1', 40']$ angular range with tomography Heymans13. The values of the cosmological parameters for the theoretical curves are given in Table\ref{['tab:CosmoParam']}. Note that the COSEBIs modes are discrete and the theory values are connected to each other for visual inspection. The errors are estimated from simulated data explained in Appendix.\ref{['app:SLICS']}. Note that the different modes are correlated (see the covariance in Fig.\ref{['fig:CovSim1bin']})
• Figure 2: Measured COSEBIs from the CFHTLenS data for 6 redshift bins using blue galaxies. The angular range $[1', 100']$ is used here. The B-modes (red circles) are shown in the upper right triangle, while the E-modes (black squares) are shown in the lower left triangle for the redshift bin pairs indicated for each panel. The theoretical values of $E_n$ are shown for the Planck (red dotted curve) and CFHTLenS+WMAP7 (blue solid curve) cosmologies (see Table\ref{['tab:CosmoParam']}). Note that the COSEBIs modes are discrete and the theory values are connected to each other for visual inspection. The errors are estimated from the mock data explained in Appendix.\ref{['app:SLICS']}. Note that the different modes are correlated as shown in Fig.\ref{['fig:CovSim6bin']}.
• Figure 3: Measured CCOSEBIs from the CFHTLenS data for 6 redshift bins using blue galaxies. The B-modes are shown as green circles. The black dashed line shows where the zero line for the B-modes lies. The measured E-modes are shown as black squares, while the theory values corresponding to the best fit values for CFHTLenS+WMAP7 and Planck (see Table\ref{['tab:CosmoParam']}) cosmologies are shown as blue solid curves and red dotted curves, respectively. Note that the CCOSEBIs modes are discrete and the theory values are connected to each other for visual inspection. The errors are estimated from simulated data explained in Appendix.\ref{['app:SLICS']}. Note that the different modes are correlated (see the covariance in Fig.\ref{['fig:CovSimCCOSEBIs']}).
• Figure 4: P-values for $\chi^2$ of B-mode compared to zero versus the number of COSEBIs modes. $n_{\rm max}$ denotes the number of COSEBIs modes from $n=1$ to $n=n_{\rm max}$. The p-value is the probability of the $\chi^2$ value being larger than the value found, assuming $B_n=0$ is the model. A very small p-value shows a poor agreement between the theory and the estimated values. We reject the null hypothesis (zero B-modes) for p-values smaller than $0.01$, which corresponds to a significance larger than $99\%$. The blue squares show the results for blue galaxies with 6 redshift bins for the largest angular scales, the light diamond belong to all galaxies with 6 redshift bins and small angular scales. Finally the grey circles show the p-values for all galaxies, a single redshift bin and the $[1',100']$ range.
• Figure 5: A comparison between two methods of finding E-COSEBIs. $E_n^{P}$ is calculated from Eq.\ref{['eq:EnFourier']}, while $E_n^{\xi}$ estimated from Eq.\ref{['eq:EnReal']}. $E_n$ with $n=1-7$ are shown here for an angular range of $[1',400']$.