Weak lensing flexion study [Elektronische Ressource] / vorgelegt von Er, Xinzhong

Er, Xinzhong

Sujets

• Astronomie

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2010
• Nombre de lectures: 29
• Langue: Deutsch
• Poids de l'ouvrage: 1 Mo

Extrait

Weak Lensing Flexion Study
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der
Rheinischen Friedrich-Wilhelms-Universit¨at Bonn
vorgelegt von
Er, Xinzhong
aus
Tianjin, China
Bonn 2010Diese Dissertation ist auf dem Hochschulschriftenserver der ULB Bonn
http://hss.ulb.uni-bonn.de/diss online
elektronisch publiziert. Das Erscheinungsjahr ist 2010.
Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at der
Rheinischen Friedrich-Wilhelms-Universit¨at Bonn
1. Referent: Prof. Dr. Peter Schneider
2. Referent: Prof. Dr. Cristiano Porciani
Tag der Promotion: 23 August 2010Contents
0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Cosmological Standard Model 3
1.1 The expansion of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Spacetime geometry and the Einstein equation . . . . . . . . . . . . . 4
1.1.2 The Robertson-Walker metric and the Friedmann equations . . . . . . 5
1.1.3 The Cosmological redshift . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Luminosity distance and angular diameter distance . . . . . . . . . . . 7
1.2 Early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Brief thermal history . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Recombination and Reionization . . . . . . . . . . . . . . . . . . . . . 12
1.3 Large-Scale Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Linear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Correlation function and power spectrum . . . . . . . . . . . . . . . . 17
1.3.4 The bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.5 Non-linear evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.6 The substructure problem . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Gravitational Lensing 21
2.1 Basic lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Deflection angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.2 Lensing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Image distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 The relation between convergence and shear . . . . . . . . . . . . . . . 25
2.2.2 Shear measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.3 Mass-sheet degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Cosmic shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Light propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 The E- and B- mode of the shear field . . . . . . . . . . . . . . . . . . 29
2.3.3 Cosmic shear statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Galaxy-Galaxy lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Weak Lensing Flexion 35
3.1 Lensing Flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Higher-order image distortions . . . . . . . . . . . . . . . . . . . . . . 35
iiiiv CONTENTS
3.1.2 Complex flexion notation . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.3 Reduced flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 E/B mode flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 The axially-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Critical curves and caustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Zero discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Non-zero discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Flexion measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Shapelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Brightness moments/ HOLICs . . . . . . . . . . . . . . . . . . . . . . 48
3.4.3 Reduced shear and flexion estimates . . . . . . . . . . . . . . . . . . . 51
3.4.4 Accuracy of brightness moment estimators . . . . . . . . . . . . . . . 53
4 Mass reconstruction of galaxy clusters 57
4.1 Mass reconstruction by flexion . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Kaiser-Squires inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.2 Finite-field inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 Mass reconstruction by shear and flexion . . . . . . . . . . . . . . . . . . . . . 59
24.2.1 Theχ -function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.2 Grid point potential field . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.3 The linearization of the equations . . . . . . . . . . . . . . . . . . . . 62
4.2.4 Numerical test with NIS toy model . . . . . . . . . . . . . . . . . . . . 64
4.2.5 Simulated cluster data . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.6 Reconstructed κ map . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Galaxy-Galaxy Lensing 71
5.1 Circular Halo profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.1 Singular Isothermal Sphere . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.2 Nonsingular Isothermal Sphere . . . . . . . . . . . . . . . . . . . . . . 73
5.1.3 Navarro-Frenk-White density profile . . . . . . . . . . . . . . . . . . . 73
5.2 Radial and Tangential Flexion. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Elliptical Halos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 Numerical test with NIE model . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Aperture flexion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.1 Weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.4.2 Mock data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.3 Aperture flexion from mock data . . . . . . . . . . . . . . . . . . . . . 81
6 Summary and Outlook 87
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A Higher-order KSB 91
A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.2 Higher-order KSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93CONTENTS v
obs isoA.2.1 FromI quantitiestoI quantities-the“smear”correctionincluding
centroid shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
iso 0ˆA.2.2 From I quantities to I quantities - the isotropic correction . . . . . 99
B The matrix C 103vi CONTENTS0.1. INTRODUCTION 1
0.1 Introduction
Asearlyasthousandsofyearsago,peoplealreadystartedstudyingtheworldofheavenandthe
universe. It is a long history involving science, philosophy, and religion, like that the ancient
Chinese philosophers term the universe by ‘Taiji’, which means ‘supreme ultimate’. Only in
recent decades, people gained solid knowledge of physics, which allowed to really explore the
universe in a scientific way. In the 1920s, Hubble found that galaxies are receding, which
is an evidence for an expansion of the universe, and is considered as the first cosmological
observation. Since then, the understanding of the universe has changed and grown rapidly
with theprogressof bothphysical theory andtelescope technology. Ourunderstandingof the
universe is contained in some model of cosmology. The one currently popular is the ΛCDM
cosmology, which is favored by most cosmological observations, such as large volume galaxy
survey(SDSS)(Tegmarketal.2004), cosmicmicrowavebackground(WMAP)(Komatsuetal.
2009)andcosmicshear(Fuetal.2008). Thesemeasurementsareaccurateenoughtoconstrain
the parameterizations of the cosmological model. With these parameters, some properties of
the universe on large-scales can be established, e.g. how the structure in the universe formed
or whether the universe is flat.
Despite these successes, we have little knowledge about the fundamental physics of the
universe. In the ΛCDM model, about 70% of the matter-energy in the universe consists of
Dark Energy and 25% of Cold Dark Matter, whose nature are still unknown. Furthermore,
the initial conditions for structure formation are given by some inflation scenarios, whose
predictions still need strict confirmation. Finally structure formation in the universe is not
fullyunderstoodonsmallscales (smallerthantheMpcscale), wherenon-linearprocessesplay
an important role. Precise measurements of the matter distribution in the universe will help
to understand some of these issues.
Weak gravitational lensing provides an excellent tool to test cosmological models. The
light of distant galaxies is deflected by the tidal gravitational field of the intervening matter
along the line of sight to the observer. The distorted imag