[Paper Review] Analysis of two-point statistics of cosmic shear: I. Estimators and covariances

Petra Schneider, Ludovic Van Waerbeke+2 more|ArXiv.org|Jun 12, 2002

Galaxies: Formation, Evolution, Phenomena42 references115 citations

TL;DR

This paper derives exact and approximate expressions for the covariance matrix of cosmic shear two-point correlation functions, enabling accurate estimation of cosmological parameters from weak lensing surveys. It introduces estimators for shear correlation functions, aperture mass dispersion, and power spectrum band powers, showing that band powers are weakly correlated and that cosmic shear can tightly constrain σ₈, Ωₘ, and Γ, especially with CMB priors.

ABSTRACT

We derive in this paper expressions for the covariance matrix of the cosmic shear two-point correlation functions which are readily applied to any survey geometry. Furthermore, we consider the more special case of a simple survey geometry which allows us to obtain approximations for the covariance matrix in terms of integrals which are readily evaluated numerically. These results are then used to study the covariance of the aperture mass dispersion which has been employed earlier in quantitative cosmic shear analyses. We show that the aperture mass dispersion, measured at two different angular scales, quickly decorrelates with the ratio of the scales. Inverting the relation between the shear two-point correlation functions and the power spectrum of the underlying projected matter distribution, we construct estimators for the power spectrum and for the band powers, and show that they yields accurate approximations; in particular, the correlation between band powers at different wave numbers is quite weak. The covariance matrix of the shear correlation function is then used to investigate the expected accuracy of cosmological parameter estimates from cosmic shear surveys. Depending on the use of prior information, e.g. from CMB measurements, cosmic shear can yield very accurate determinations of several cosmological parameters, in particular the normalization $σ_8$ of the power spectrum of the matter distribution, the matter density parameter $Ω_{ m m}$, and the shape parameter $Γ$.

Motivation & Objective

• To derive unbiased estimators for cosmic shear two-point correlation functions from observational data with complex geometry.
• To compute the full covariance matrix of these estimators, accounting for both measurement noise and cosmic variance.
• To develop approximations for the covariance under simplified survey geometries (e.g., compact, single region) using numerical integrals.
• To derive the covariance of derived statistics such as the aperture mass dispersion and power spectrum band powers.
• To evaluate the expected accuracy of cosmological parameter constraints (σ₈, Ωₘ, Γ, zₛ) from cosmic shear surveys using likelihood and marginalization techniques.

Proposed method

• Derives unbiased estimators for the two-point shear correlation functions ξ₊(ϑ) and ξ₋(ϑ) using galaxy pair counts in a survey.
• Expresses the full covariance matrix of the correlation function estimators in terms of galaxy positions and intrinsic ellipticity dispersion.
• Applies the four-point function factorization approximation (⟨γγγγ⟩ ≈ ⟨γγ⟩⟨γγ⟩) to reduce the four-point correlation to products of two-point functions.
• Derives ensemble-averaged covariance expressions for a compact survey region by averaging over all angular separations under the assumption ϑ ≪ √A.
• Uses numerical integration to evaluate the resulting integrals for the ensemble-averaged covariance matrix in compact survey geometries.
• Constructs estimators for the power spectrum Pₖ(ℓ) and its band powers from the measured ξ₊ and ξ₋, showing weak cross-correlation between band powers.

Experimental results

Research questions

• RQ1 How can the covariance matrix of cosmic shear two-point correlation functions be accurately computed for arbitrary survey geometries?
• RQ2 What are the dominant contributions to the covariance—measurement noise or cosmic variance—and how do they depend on survey geometry?
• RQ3 How do the estimators for aperture mass dispersion and power spectrum band powers behave in terms of bias and correlation?
• RQ4 To what extent can cosmic shear surveys constrain cosmological parameters like σ₈, Ωₘ, Γ, and source redshift zₛ, especially with and without priors?
• RQ5 How quickly do shear correlation function estimates decorrelate across different angular scales, and what does this imply for parameter degeneracy?

Key findings

• The covariance of the shear correlation function estimators depends on the number of galaxy pairs, survey area, geometry, and intrinsic ellipticity dispersion.
• Estimates of ξ₋(ϑ) decorrelate rapidly with scale, becoming essentially uncorrelated for angular separations differing by more than a factor of two.
• Estimates of ξ₊(ϑ) remain correlated over much larger angular scales, indicating stronger scale-dependent signal coherence.
• The cross-correlation between ξ₊(ϑ₁) and ξ₋(ϑ₂) is significant when ϑ₁ ≤ ϑ₂, reflecting the different filtering properties of the correlation functions.
• The power spectrum band power estimator constructed from ξ₊ and ξ₋ yields accurate approximations with weak correlations between band powers at different wave numbers.
• Cosmic shear surveys can provide tight constraints on σ₈, Ωₘ, and Γ, especially when combined with CMB priors, with the Ωₘ–σ₈ pair showing the strongest constraints and least degeneracy.