Cosmic Shear Tomography and Efficient Data Compression using COSEBIs

TL;DR

The paper evaluates COSEBIs as a compact, E-/B-mode separable statistic for cosmic shear data in a seven-parameter, tomographic framework, using Fisher analysis to quantify information content. It demonstrates that a small set of COSEBI modes, especially the logarithmic version, saturates cosmological information and that tomography significantly enhances parameter constraints, with large surveys providing the strongest gains. By formulating covariances directly from the power spectrum, it achieves efficient data compression and faster analyses, while highlighting the advantages of Log-COSEBIs over Lin-COSEBIs. The results support COSEBIs as a practical tool for current and future cosmic shear analyses, enabling robust parameter estimation with manageable computational requirements.

Abstract

Context. Gravitational lensing is one of the leading tools in understanding the dark side of the Universe. The need for accurate, efficient and effective methods which are able to extract this information along with other cosmological parameters from cosmic shear data is ever growing. COSEBIs, Complete Orthogonal Sets of E-/B-Integrals, is a recently developed statistical measure that encompasses the complete E-/B-mode separable information contained in the shear correlation functions measured on a finite angular range. Aims. The aim of the present work is to test the properties of this newly developed statistics for a higher-dimensional parameter space and to generalize and test it for shear tomography. Methods. We use Fisher analysis to study the effectiveness of COSEBIs. We show our results in terms of figure-of-merit quantities, based on Fisher matrices. Results. We find that a relatively small number of COSEBIs modes is always enough to saturate to the maximum information level. This number is always smaller for 'logarithmic COSEBIs' than for 'linear COSEBIs', and also depends on the number of redshift bins, the number and choice of cosmological parameters, as well as the survey characteristics. Conclusions. COSEBIs provide a very compact way of analyzing cosmic shear data, i.e., all the E-/B-mode separable second-order statistical information in the data is reduced to a small number of COSEBIs modes. Furthermore, with this method the arbitrariness in data binning is no longer an issue since the COSEBIs modes are discrete. Finally, the small number of modes also implies that covariances, and their inverse, are much more conveniently obtainable, e.g., from numerical simulations, than for the shear correlation functions themselves.

Figures (15)

• Figure 1: The weight functions $W_n^\mathrm{Lin}(\ell)$ are the Hankel transforms of $T_{\pm}^\mathrm{Lin}(\vartheta)$ as in Eq.\ref{['Wn']}. In the blow-ups, the two modes of oscillation for each $W_n^\mathrm{Lin}$ can be seen, the lower frequency mode and the higher frequency mode which are inversely proportional to $\vartheta_{\mathrm{min}}$ and $\vartheta_{\mathrm{max}}$, respectively. The overall amplitude of the oscillations strongly depends on $n$ and $\vartheta_{\mathrm{max}}$.
• Figure 2: The weight functions $W_n^\mathrm{Log}(\ell)$ are the Hankel transformation of $T_{\pm}^\mathrm{Log}(\vartheta)$ as in Eq.\ref{['Wn']}. Similar to the $W_n^\mathrm{Lin}$, the position of the first peak depends mainly on $\vartheta_\mathrm{max}$ and is rather insensitive to $\vartheta_\mathrm{min}$. The difference between the two sets of linear and logarithmic function can be seen most prominently in the blow-ups; the lower frequency oscillations are more pronounced in this case.
• Figure 3: The overall source redshift probability distribution of source galaxies assumed for the two surveys. LS has a deeper source distribution compared to MS.
• Figure 4: The absolute value of the derivative of the convergence power spectrum with respect to $\Omega_\mathrm{m}$. Both of the curves rely on a five point stencil method where 4 nearby points have to be evaluated. The solid curve is drawn assuming all parameters are fixed except $\Omega_\mathrm{m}$ and $\Gamma$, in contrast to the dotted curve where instead of $\Gamma$, $h$ or $\Omega_\mathrm{b}$ are variable.
• Figure 5: A 3D representation of the non-tomographic covariance of 15 E-mode COSEBIs for an angular range of $[1',400']$, for MS parameters. The $x$- and $y$- axes correspond to the elements of the covariance matrix, and the value of the vertical axis shows the value of the covariance of the corresponding element. A contour representation of the covariance is shown for each plot at its base.