The Skewness of the Aperture Mass Statistic

TL;DR

This paper addresses measuring the skewness of the aperture mass, $\langle M_{ m ap}^3\rangle$, in weak lensing while remaining robust to survey masking. It derives practical formulae that express higher-order aperture-mass moments in terms of the three-point shear correlation function, aided by analytic kernels $T_0$ and $T_1$, and provides efficient tree-based algorithms (two-point, three-point, and a faster PairCell approach) for large data sets. The methods are validated on simulated data and applied to the CTIO survey, yielding a marginal positive skewness at about $2\sigma$ but limited by noise and systematics for cosmological constraints. The work suggests that upcoming wide and deep surveys (e.g., CFLS, SNAP, LSST) will achieve sufficient signal-to-noise to exploit non-Gaussian lensing measurements for cosmology. Key mathematical relations include $\langle M_{ m ap}^2\rangle(R) = \frac{1}{2} \int \frac{s ds}{R^2}[\xi_+(s) T_+(s/R) + \xi_-(s) T_-(s/R)]$ and $\langle M_{ m ap}^3\rangle(R) = \frac{1}{4}\Re\left(3\langle M^2 M^*\rangle(R) + \langle M^3\rangle(R)\right)$, with context provided by the three-point components $\Gamma_i$ and kernels $T_0,T_1$ that encode triangle geometry.$

Abstract

We present simple formulae for calculating the skewness and kurtosis of the aperture mass statistic for weak lensing surveys which is insensitive to masking effects of survey geometry or variable survey depth. The calculation is the higher order analog of the formula given by Schneider et al (2002) which has been used to compute the variance of the aperture mass from several lensing surveys. As our formula requires the three-point shear correlation function, we also present an efficient tree-based algorithm for measuring it. We show how our algorithm would scale in computing time and memory usage for future lensing surveys. Finally, we apply the procedure to our CTIO survey data, originally described in Jarvis et al (2003). We find that the skewness is positive (inconsistent with zero) at the 2 sigma level. However, the signal is too noisy from this data to usefully constrain cosmology.

Figures (8)

• Figure 1: Graphical representation of the $q$ parameters used in the formulae for $T_0$ and $T_1$. These vectors are used as complex numbers in the formulae.
• Figure 2: The absolute magnitude of the functions $T_0$ and $T_1$ for equilateral triangles as a function of the side length of the triangle, $s$. The solid green curve is $|T_0|$, and the dashed blue curve is $|T_1|$.
• Figure 3: A sample calculation step in the two-point algorithm. At first the two Cells are too large compared to the distance between them, so they need to be split up. After the splits, there are four pairs of subcells to be considered. Three of the pairs (marked with green solid lines) satisfy Equation (\ref{['soverd']}), and thus can be computed directly. The fourth (marked with a red dashed line) needs to be split again.
• Figure 4: The geometry of the triangle in the discussion of the three-point algorithm. We use the slightly non-intuitive convention that $d_1 > d_2 > d_3$ (rather than the other way around). This is so we agree with Fig. \ref{['qfig']} with s as the smallest side, as desired for Equations \ref{['mcbform']} and \ref{['msqmcform']}.
• Figure 5: The geometry of the PairCell's for the improved three-point algorithm. The pairs on the left side of the figure all have midpoints which are near each other, and have cells at roughly the same separation. Thus, they all fall into a single PairCell. The dots represent the midpoints of each pair. The size of the PairCell, $s_{12}$, is based on the scatter of these midpoints. The distance from the PairCell to another Cell is then the median of the corresponding triangles, $d_m$.