Karhunen-Loeve Analysis for Weak Gravitational Lensing

7.3 MB·English

by  Jacob T Vanderplas

(cid:13)c Copyright 2012
Jacob T. Vanderplas
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Karhunen-Lo`eve Analysis for Weak Gravitational Lensing
Jacob T. Vanderplas
A dissertation
submitted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
University of Washington
2012
Reading Committee:
Andrew Connolly, Chair
Bhuvnesh Jain
Andrew Becker
Program Authorized to Offer Degree:
Department of Astronomy

University of Washington
Abstract
Karhunen-Lo`eve Analysis for Weak Gravitational Lensing
Jacob T. Vanderplas
Chair of the Supervisory Committee:
Professor Andrew Connolly
Department of Astronomy
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-Lo`eve (KL) analysis to speed and improve the comparison
of weak lensing shear catalogs to theory in order to constrain cosmological parameters in
currentandfuturelensingsurveys. Thisworkaddressesthreerelatedaspectsofweaklensing
analysis:
Three-dimensional Tomographic Mapping: (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
toevaluatetheefficacyoflinearreconstructiontechniques. WefindthattheKL-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.
Shear Peak Statistics with Incomplete Data (Based on work published in Vander-
Plasetal.,2012)WeexploretheuseofKLeigenmodesforinterpolationacrossmasked

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 de-
rived 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.
Two-point parameter estimation with KL modes The power spectrum of the ob-
served shear can yield powerful cosmological constraints. Incomplete survey sky cov-
erage, however, canleadtomixingofpowerbetweenFouriermodes, andobfuscatethe
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.

TABLE OF CONTENTS
Page
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Chapter 1: Brief Introduction to Cosmology . . . . . . . . . . . . . . . . . . . . . 1
1.1 FLRW Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The Friedmann Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Time Dilation and Redshift . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Evolving Equation of State . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Hubble Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Cosmological Distance Measures . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Standard Candles: Cosmology via Luminosity Distance . . . . . . . . . . . . 12
1.5 The Growth of Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5.1 Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5.2 Perturbation Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.3 Matter Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.4 Putting it all together . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.1 Simplifying Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6.2 Lensing Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.3 Continuous Mass Distribution . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Weak Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.7.1 Mapping with Weak Lensing . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.2 Power Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Chapter 2: Introduction to Karhunen-Lo`eve Analysis . . . . . . . . . . . . . . . . 32
2.1 Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Basis function decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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2.2.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Generalizing Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . 35
2.3 Karhunen-Lo`eve Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Derivation of Karhunen-Lo`eve theorem. . . . . . . . . . . . . . . . . . 36
2.3.2 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.3 Partial Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.4 KL in the presence of noise . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.5 Karhunen-Lo`eve: theory to practice . . . . . . . . . . . . . . . . . . . 42
2.3.6 KL with missing data . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4 Karhunen-Lo`eve Analysis and Bayesian Inference . . . . . . . . . . . . . . . . 47
2.5 Karhunen-Lo`eve Analysis of Shear . . . . . . . . . . . . . . . . . . . . . . . . 48
Chapter 3: 3D weak lensing maps with KL . . . . . . . . . . . . . . . . . . . . . . 50
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 Linear Mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.2 KL Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Singular Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.2 Evaluation of the SVD Estimator . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Comparison of Estimators . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.4 Noise Properties of Line-of-Sight Modes . . . . . . . . . . . . . . . . . 59
3.3.5 Reconstruction of a Realistic Field . . . . . . . . . . . . . . . . . . . . 60
3.3.6 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Chapter 4: Shear Peak Statistics with KL . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Karhunen-Lo`eve Analysis of Shear . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.1 KL Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 KL in the Presence of Noise . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2.3 Computing the Shear Correlation Matrix . . . . . . . . . . . . . . . . 74
4.2.4 Which Shear Correlation? . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.5 Interpolation using KL Modes . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Testing KL Reconstructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 KL Decomposition of a Single Field . . . . . . . . . . . . . . . . . . . 79
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