Gravitational Lensing in Relativistic Modified Newtonian Dynamics

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研究生：	田雍 Yong Tian
論文名稱：	Gravitational Lensing in Relativistic Modified Newtonian Dynamics Gravitational Lensing in Relativistic Modified Newtonian Dynamics
指導教授：	高仲明 Chung-Ming Ko 聶斯特 James M. Nester
口試委員:
學位類別：	博士 Doctor
系所名稱：	理學院 - 物理學系 Department of Physics
論文出版年：	2014
畢業學年度：	102
語文別：	英文
論文頁數：	112
中文關鍵詞：	重力、重力透鏡、暗物質、修改牛頓力學、重力時間延遲
外文關鍵詞：	Gravity, Gravitational Lensing, Dark Matter, Modified Newtonian Dynamics, Gravitational Time Delay
相關次數：	點閱：21 下載：0
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暗物質問題，或是正準確的說消失質量問題，長久以來一直是天文物理上最重要的問題。我們在這份工作的第一章，從銀河系尺度到宇宙學回顧暗物質問題。接著，我們在第二章著重暗物質的替代方案“修改牛頓重力理論”，其將質量消失問題看做是加速度異常。
第三章，我們介紹了“修改牛頓重力理論”的相對論性重力理論的版本並給出自洽的微中子宇宙學。我們的工作特別著重在相對論性重力理論的版本中的重力透鏡和重力時間延遲的效應上。第四章，我們推導出在相對論性重力理論的版本中的重力透鏡公式並且在第五章測試了這個公式在強重力透鏡的效應。雖然結果是初步的只針對球對稱的情況，做為透鏡的橢圓銀河系質量和重力時間延遲所得出的哈伯常數都很正確的符合目前的觀測結果，並與相對論性重力理論的架構自洽。甚至，從動力學得出的透鏡橢圓銀河系質量和重力透鏡所得出的也有一致性。
第六章，我們總結了一些結果並給出我們的看法，同時點出未來可以進行的工作，希望能幫助未來在這個主題上的學習。

The dark matter, or more accurately “missing mass”, problem could well be the most important and most long standing issue in astrophysics. In this work, we review this problem from the galaxies scale to the cosmology in Chapter I. Then, we focus on the alternative solution “Modified Newtonian Dynamics” (MOND) which interprets “missing mass” as “acceleration discrepancy” in Chapter II.
In Chapter III, we introduce relativistic gravity theories in MOND and give the consistent neutrino cosmology. Our work focuses especially on gravitational lensing and time delay within the relativistic version of MOND. We derive the lensing formulation in relativistic MOND in Chapter VI and test it on strong lensing cases in Chapter V. Although the results are preliminary for the spherically symmetric case, the lens mass of an elliptical galaxy in lensing and the Hubble constant in time delay are still consistent within the relativistic MOND paradigm. Moreover, the masses of lens galaxies estimated from dynamics and from gravitational lensing are also consistent in our study.
In Chapter VI, we conclude with some comments and point out directions for future work which should be helpful for further study of this topic.

Introduction 1
1 Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Flat Rotational Curves . . . . . . . . . . . . . . . . . . 2
1.2 Tully-Fisher Relation . . . . . . . . . . . . . . . . . . . 3
1.3 Insufficient Dark Matter in Elliptical Galaxies . . . . . 4
1.4 Gravitational Lensing and Time Delay . . . . . . . . . 5
2 Clusters of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 X-ray Emission . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Weak Lensing on Bullet Cluster . . . . . . . . . . . . . 8
2.3 Gravitational Redshift . . . . . . . . . . . . . . . . . . 10
3 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Big Bang Nucleosynthesis . . . . . . . . . . . . . . . . 13
3.2 Hubble Diagram for High Redshift . . . . . . . . . . . 14
3.3 Cosmic Microwave Background Radiation . . . . . . . 16
Alternative to Dark Matter 21
1 MOdified Newtonian Dynamics . . . . . . . . . . . . . . . . . 21
1.1 Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 AQUAL: non-Relativistic Field Reformulation of MOND 23
1.3 Interpolation Function . . . . . . . . . . . . . . . . . . 26
1.4 Dynamics in Accelerating Universe . . . . . . . . . . . 27
2 Solutions in MOND . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1 Spiral Galaxies in MOND . . . . . . . . . . . . . . . . 28
2.2 Baryonic Tully-Fisher Relation in MOND . . . . . . . 29
2.3 Elliptical Galaxies in MOND . . . . . . . . . . . . . . . 30
2.4 Clusters of Galaxies in MOND . . . . . . . . . . . . . . 30
2.5 Gravitational Redshift in MOND . . . . . . . . . . . . 33
Relativistic MOND 39
1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.1 Relativistic AQUAL . . . . . . . . . . . . . . . . . . . 39
1
CONTENTS
1.2 Disformal Related Metric . . . . . . . . . . . . . . . . . 40
2 TeVe S:Tensor-Vector-Scalar Theory . . . . . . . . . . . . . . 42
2.1 Fields and Actions . . . . . . . . . . . . . . . . . . . . 42
2.2 Non-Relativistic MOND Limit . . . . . . . . . . . . . . 43
2.3 The MOND Limit: Spherical Symmetry . . . . . . . . . 44
3 GEA: Generalized Einstein-Aether Theories . . . . . . . . . . . 46
4 BiMOND: Bimetric MOND Theories . . . . . . . . . . . . . . 47
5 Neutrino Cosmological Model . . . . . . . . . . . . . . . . . . 48
5.1 Active Neutrino . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Sterile Neutrino . . . . . . . . . . . . . . . . . . . . . . 50
Theoretic Aspects 55
1 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . . 55
2 Lensing and Time Delay Equation . . . . . . . . . . . . . . . . 57
2.1 The General Static Isotropic Metric . . . . . . . . . . . 57
2.2 General Equation of Motion . . . . . . . . . . . . . . . 57
2.3 The Schwarzschild Solution . . . . . . . . . . . . . . . 58
2.4 Deflection Angle . . . . . . . . . . . . . . . . . . . . . 59
2.5 Lens and Time Delay Equation . . . . . . . . . . . . . 60
3 Mass Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1 Point Mass Model in Bekenstein Form . . . . . . . . . 62
3.2 Hernquist Model in Bekenstein Form . . . . . . . . . . 65
3.3 Deep MOND Limit and MOND Lensing Prediction . . 67
Observational Aspects 71
1 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . 71
1.1 Quasar Strong Lensing . . . . . . . . . . . . . . . . . . 71
1.2 Strong Lensing and Dynamical Mass in Elliptical Galaxies
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2 Gravitational Time Delay . . . . . . . . . . . . . . . . . . . . 79
2.1 Shapiro Time Delay . . . . . . . . . . . . . . . . . . . . 79
2.2 Quasar Time Delay . . . . . . . . . . . . . . . . . . . . 79
2.3 Weak Lensing in the Bullet Cluster . . . . . . . . . . . 83
Conclusion and Outlook 89
1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.1 Prospect of Gravitational Lensing in Relativistic MOND 91
2.2 Elliptical Galaxies in MOND and IMF Stellar Mass . . 92
2.3 Fundamental Concept Behind MOND . . . . . . . . . . 93
                                

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