Karhunen-Loeve Analysis for Weak Gravitational Lensing
VanderPlas, Jacob T.
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In the past decade, weak gravitational lensing has become an important tool in the study of the universe at the largest scale, giving insights into the distribution of dark matter, the expansion of the universe, and the nature of dark energy. This thesis research explores several applications of Karhunen-Loeve (KL) analysis to speed and improve the comparison of weak lensing shear catalogs to theory in order to constrain cosmological parameters in current and future lensing surveys. This work addresses three related aspects of weak lensing analysis: <bold>Three-dimensional Tomographic Mapping:</bold> (Based on work published in Vanderplas et al 2011) We explore a new fast approach to three-dimensional mass mapping in weak lensing surveys. The KL approach uses a KL-based filtering of the shear signal to reconstruct mass structures on the line-of-sight, and provides a unified framework to evaluate the efficacy of linear reconstruction techniques. We find that the KL-based filtering leads to near-optimal angular resolution, and computation times which are faster than previous approaches. We also use the KL formalism to show that linear non-parametric reconstruction methods are fundamentally limited in their ability to resolve lens redshifts. <bold>Shear Peak Statistics with Incomplete Data:</bold> (Based on work published in Vanderplas et al 2012) We explore the use of KL eigenmodes for interpolation across masked regions in observed shear maps. Mass mapping is an inherently non-local calculation, meaning gaps in the data can have a significant effect on the properties of the derived mass map. Our KL mapping procedure leads to improvements in the recovery of detailed statistics of peaks in the mass map, which holds promise of improved cosmological constraints based on such studies. <bold>Two-point parameter estimation with KL modes:</bold> The power spectrum of the observed shear can yield powerful cosmological constraints. Incomplete survey sky coverage, however, can lead to mixing of power between Fourier modes, and obfuscate the cosmologically sensitive signal. We show that KL can be used to derive an alternate orthonormal basis for the problem which avoids mode-mixing and allows a convenient formalism for cosmological likelihood computations. Cosmological constraints derived using this method are shown to be competitive with those from the more conventional correlation function approach. We also discuss several aspects of the KL approach which will allow improved handling of correlated errors and redshift information in future surveys.
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