Analysis of two-point statistics of cosmic shear: I. Estimators and covariances

TL;DR

This paper develops practical, survey-geometry–dependent expressions for the covariance of cosmic shear two-point statistics, including estimators for the shear correlation functions, aperture-mass measures, and the projected matter power spectrum. By assuming Gaussian shear, it derives explicit covariances, analyzes the impact of survey geometry and shot noise, and shows that aperture-mass estimators decorrelate rapidly with scale. It then demonstrates how these covariances enable reliable constraints on cosmological parameters, notably yielding strong constraints on $σ_8$ and informative degeneracy directions among parameters, with methods for constructing band powers and a simple power-spectrum estimator. The results provide a foundation for optimizing future cosmic shear surveys and for robustly extracting cosmological information from two-point statistics.

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 $Ω_{\rm m}$, and the shape parameter $Γ$.

Figures (8)

• Figure 1: The correlation functions. In the left panel, we have plotted the covariance matrices ${\rm Cov}_{++}'({\vartheta}_1,{\vartheta}_2)$ (thick solid curves) and ${\rm Cov}_{--}'({\vartheta}_1,{\vartheta}_2)$, i.e. the covariance matrices with the shot-noise term removed. For ${\rm Cov}_{++}'({\vartheta}_1,{\vartheta}_2)$, the contours are linearly spaced, with the lowest value at $10^{-9}$ (outer-most contour) and highest value $9\times 10^{-9}$ for small ${\vartheta}_1$, ${\vartheta}_2$. For ${\rm Cov}_{--}'({\vartheta}_1,{\vartheta}_2)$, contours are logarithmically spaced, with consecutive contours differing by a factor 1.5. The solid contours display positive values of ${\rm Cov}_{--}'({\vartheta}_1,{\vartheta}_2)$, starting from $10^{-14}$, with the maximum value of $\sim 3\times 10^{-9}$ in the upper right corner, and dotted contours show negative values of ${\rm Cov}_{--}'({\vartheta}_1,{\vartheta}_2)$, starting at $-10^{-15}$. In the right panel, ${\rm Cov}_{+-}({\vartheta}_1,{\vartheta}_2)$ is shown, again with logarithmically spaced contours differing by a factor of 1.5. Solid contours are for positive values of ${\rm Cov}_{+-}({\vartheta}_1,{\vartheta}_2)$, starting at $10^{-14}$, negative values are shown by dotted contours, starting at $-10^{-13}$.
• Figure 2: Left panel: The square root of the variances $\sqrt{{\rm Var}(\hat{\xi}_+;{\vartheta})}$ and $\sqrt{{\rm Var}(\hat{\xi}_-;{\vartheta})}$ shown as dotted and long-dashed curves, together with the correlation functions $\xi_+({\vartheta})$ and $\xi_-({\vartheta})$ as solid and short-dashed curves, respectively. The model parameters are as described in the text; in particular, a fiducial value of the survey area of $1\,{\rm deg}^2$ has been taken. For the diagonal part of the covariance matrix, we have assumed a relative bin size of $\Delta{\vartheta}/{\vartheta}=0.1$. For small ${\vartheta}$, the variance behaves as ${\vartheta}^{-1}$, as it is dominated by the noise from the intrinsic ellipticity of the source galaxies, i.e. the term $D$ (\ref{['eq:Ddef']}), whereas for larger values of ${\vartheta}$, the main contribution comes from cosmic variance. Right panel: The correlation coefficient $r_{\rm corr}$, as defined in (\ref{['eq:rcorrxi']}), as a function of ${\vartheta}_2$, for various values of ${\vartheta}_1$. Solid curves show $r_{\rm corr}(\hat{\xi}_+)$, dashed curves show $r_{\rm corr}(\hat{\xi}_-)$. The value of ${\vartheta}_1$ corresponding to each curve can be read off from the point where a curve attains the value $r_{\rm corr}=1$.
• Figure 3: Left panel: The square-root of the autovariance of $\mathcal{M}$ as a function of angular scale. The long-dashed and dash-dotted curves show the minimum variance, i.e. in the absence of a shear correlation; this variance is due solely to the intrinsic ellipticity of source galaxies. Curves are shown for $K_+=0, 1/2, 1$, where the minimum variance is the same for $K_+=0$ and $1$. The solid, dashed and dotted curves show the variance in the presence of a cosmic shear; also here, the cases $K_+=0$ and $K_-=1$ are nearly the same, and the variance is smallest for $K_+=1/2$. For comparison, $\left\langle M_{\rm ap}^2 \right\rangle(\theta)$ is plotted as thick solid curve. As for the other figures shown before, our standard set of parameters $A=1\,{\rm deg}^2$, $n=30\,{\rm arcmin}^{-2}$ and $\sigma_{\epsilon}=0.3$ has been used; the variance scales as $A^{-1}$. Right panel: The correlation coefficient $r_{\rm corr}(\mathcal{M};\theta_1,\theta_2)$ of the covariance of the estimator $\mathcal{M}$ is plotted as a function of $\theta_2$, for various values of $\theta_1$; the values of $\theta_1$ can be localized as those points where the correlation function attains the value unity. The solid curves are for $K_+=0$, i.e. when only the correlation function $\xi_-$ is used in the estimate of $\mathcal{M}$, the dotted curves are for $K_+=1/2$, and the dashed curves for $K_+=1$. The width of all three families of curves is very similar and (in logarithmic terms) basically independent of $\theta_1$. The $K_+=0$ curves do not develop a tail of anticorrelation, as is the case for $K_+=1$ (and therefore also for $K_+=1/2$). Hence, whereas $K_+=1/2$ yields the smallest variance of the estimator $\mathcal{M}$, it leads to a small but long-range correlation between different angular scales
• Figure 4: The thick solid line displays the dimensionless projected power spectrum $\ell^2 {\cal P}_\kappa(\ell)$, whereas the other two curves show the 'observed' power spectrum, as defined in (\ref{['eq:Pobs']}). The dotted curve is for $K_+=0$, i.e. only $\xi_-$ enters the determination of the observed power spectrum in this case; the dashed curve is for $K_+=1$. In this plot is was assumed that the correlation functions are known between $\theta_{\rm min}=6"$ and $\theta_{\rm min}=2^\circ$
• Figure 5: The large panel shows the estimates of the band power, shown as horizontal bars whose length indicates the bins used. The error bar on each bin shows the square root of the autovariance of the band power, and the solid curve is the underlying power spectrum, $\ell^2 P_\kappa(\ell)$. For this figure, we have assumed that the correlation functions are measured for $6"\le{\vartheta}\le 2^\circ$, from a survey of $A=25\,{\deg}^2$. The inset figure shows the correlation coefficient between the 13 different bins, where the triangles indicate the center $\bar{\ell}$ of each bin. One sees that the bands are very little correlated, except for the three bins with smallest $\ell$; in fact, the first three band power estimates are fully correlated. This explains why the band-power estimator yields reasonable results even for $\ell < 2\pi/\theta_{\rm max}\sim 180$ -- this is just a coincidence.