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The Galaxy–Dark Matter Connection

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The Galaxy–Dark Matter Connection Master’s Thesis The galaxy–dark matter connection by Tobias Baldauf University of Zurich [email protected] supervised by Prof. Dr. Uroš Seljak Institute for Theoretical Physics University of Zurich Physics and Astronomy Department University of California, Berkeley and Dr. Robert E. Smith Institute for Theoretical Physics University of Zurich Zurich, February 2009 Abstract We use dissipationless N-body simulations to investigate the correlation between dark matter and galaxy distribution in the Universe. The dark matter haloes identified in the N-body simulations are populated with galaxies according to the Halo Occupation Distribution. We then use galaxy-clustering and galaxy-galaxy lensing measurements for Luminous Red Galaxies from the Sloan Digital Sky Survey to fit for the free parameters of the occupation statistics. In this vein we obtain realistic galaxy catalogues and use them for our examination of the cross- correlation coefficient between the LRGs and the matter. As our key result we propose and test a new method to constrain the dark matter autocorrelation from observable galaxy-galaxy lensing and galaxy clustering. Most of our conclusions are based on the projected correlation function w(R), which leads us to quantify its dependence on redshift space distortions. As a side project we implement a fast grid based method for the calculation of correlation functions and their projections from N-body simulations. To my parents Acknowledgements First of all I would like to thank Prof. Dr. Uroš Seljak for providing me with this topic and enabling me to carry out my research in his group. Guiding me through this work and giving me inspirations for new aspects of the topic helped me to develop a comprehensive picture of the subject. I would like to thank Dr. Robert Smith for the enormous support he has given to me. En- couraging me to critically reconsider some of the preliminary results and to carefully study the theoretical framework helped me to evolve towards a better physicist. I also appreciate him for hours of discussions, for bearing my infinite number of questions and for valuable comments on the manuscript of this work. I am also very grateful to Dr. Vincent Desjacques and Dr. Ilian Iliev for discussions and help with the simulations during the initial stages of the project. The galaxy clustering and galaxy-galaxy lensing data used in this thesis were kindly provided by Rachel Mandelbaum, Institute for Advanced Study, Princeton. The Institute for Theoretical Physics provided a perfect working environment, both in terms of the computing equipment as well as in terms of the nice atmosphere. Particularly I would like to thank the PHD students Nico Hamaus and Lucas Lombriser and my office mate Willy Kranz for hours of discussions about physics, programming and daily life. Special thanks to my family for enabling me my studies and for pointing out to me that there are other things in life than physics. Finally, I gratefully acknowledge support by a grant of the German Academic Foundation. Zurich, February 2009 Contents 1 Introduction 1 1.1 Historical Context . 1 1.2 Cosmic Timeline . 2 1.3 Motivation . 3 2 The Expanding Universe 7 2.1 Fundamentals of General Relativity . 7 2.2 Fundamentals of Cosmology . 8 2.3 The Nature of Redshift . 11 2.4 Age of the Universe . 11 2.5 Luminosity Measures . 12 2.6 Cosmological Distances . 13 2.7 Cosmological Parameters . 13 2.8 Dark Matter . 14 3 Linear Growth of Structures 15 3.1 Newtonian Theory of Structure Formation . 15 3.2 Lagrangian Description of Structure Formation - Zeldovich Approximation . 18 3.3 Inflation in a Nutshell . 19 3.4 Cosmological Perturbation Theory . 19 3.5 Popular Gauge Choices . 20 3.6 Perturbations from Inflation . 21 3.7 Evolution of the Primordial Spectrum . 24 4 Statistics 27 4.1 Correlation Functions . 27 4.2 Filtering of the Density Field . 28 4.3 Two-Point Probability Distribution . 28 4.4 Bias and Cross-Correlation Coefficient . 29 4.5 Gaussian Random Fields . 30 5 Gravitational Clustering 31 5.1 Spherical Collapse Model . 31 5.2 Press & Schechter Theory of Peaks . 32 5.3 Mass Dependent Halo Bias . 34 5.4 Halo Density Profiles . 35 5.5 Halo Model & Halo Occupation Distribution . 35 2 | Contents 6 Gravitational Lensing 39 6.1 Basic Principles of Gravitational Lensing . 39 6.2 Weak Lensing . 41 6.3 Galaxy-Galaxy Lensing . 42 6.4 Derivation of the Mass-Shear Relation . 44 6.5 The Lens-Galaxy Sample . 45 7 Methodology 47 7.1 The zHORIZON Simulations . 47 7.2 N-Body Codes . 48 7.3 Identification of Gravitationally Bound Objects - The Halo Finder . 50 7.4 Grid Based Analysis I - Basic Principles . 51 7.5 Projected Correlation Functions . 53 7.6 Grid Based Analysis II - Density Superposition . 55 7.7 Maximum Likelihood Parameter Estimation . 59 7.8 Generation of LRG Galaxy Catalogues . 60 7.8.1 General Considerations . 61 7.8.2 The HOD Adaption Algorithm . 61 7.8.3 Modelling of the Bright LRG Sample . 63 7.8.4 Modelling of the Faint LRG Sample . 64 7.9 Redshift Space Distortions . 65 8 Results 69 8.1 Correlation Functions from the Grid Based Analysis . 69 8.2 Generation of LRG Galaxy Catalogues . 70 8.2.1 Influence of the HOD Parameters on the Projected Correlations . 71 8.2.2 Best Fit HOD: Bright sample . 74 8.2.3 Best Fit HOD: Faint sample . 74 8.3 Cross-Correlation Coefficient . 76 8.3.1 Haloes . 76 8.3.2 Galaxies: Projected Correlation . 76 8.3.3 Galaxies: Compensated Surface Mass Density . 78 8.4 Redshift Space Distortions on Projected Correlation Functions . 83 8.5 Small Scale Truncation . 84 8.6 Influence of Satellite Galaxies . 86 8.7 NFW Versus Dark Matter . 86 8.8 Projected Correlation Functions . 88 9 Conclusions 89 9.1 Analysis Techniques . 89 9.2 Generation of LRG Galaxy Catalogues . 90 9.3 Cross-Correlation Coefficient and Dark Matter Recovery . 91 9.4 Effects Related to the Projected Correlation Functions . 92 9.5 Future Prospects and Possible Improvements . 92 A Notation and Constants 95 CHAPTER 1 Introduction This first chapter will start with a short description of the major scientific cognitions that lead to the current understanding of the Universe, before we proceed with a summary of important events in the cosmic history. Finally we will try to motivate our investigations. The organisation of the following chapters is as follows: In §2 we will lay down the fundamentals of general relativity and cosmology in a smooth homogeneous Universe, before we proceed to the origin of perturbations on the smooth background Universe in §3. The discussion in §4 will then be devoted to the statistics, frequently used to describe the cosmological random fields, especially power spectra and correlation functions. The formation of non-linear structures will be treated in §5, where we also derive the abundance of dark matter haloes and comment on the halo model of large scale structure. Gravitational lensing, especially weak lensing will be covered in §6. This chapter also provides the necessary formalism for our galaxy-galaxy lensing analysis. All this theory is then used in §7 to develop the techniques necessary for our analysis, especially for the extraction of correlation functions from the simulations and for the adaption of the occupation statistics. Finally we will describe our findings in §8 and discuss them in §9. Readers familiar with the concepts of cosmology should directly jump to §6 on Page 39, where we start to present our own achievements. 1.1 Historical Context Mankind was observing the night sky for thousands of years but it was not before the 16th and 17th century that physicists like Galileo Galilei, Johannes Kepler and Isaac Newton developed a mathematical theory of gravitation which could coherently account for the planetary orbits. This very successful theory of gravitational force turned out to be the weak field limit of general relativity, which was presented by Albert Einstein in 1915. Based on his earlier work on special relativity, he developed a geometric theory of gravity, which abandons the idea of gravity acting as a force and rather considers it as a property of a curved four dimensional spacetime. The core of the mathematical realisation of this theory are the Einstein field equations, relating curvature of spacetime to its energy content. The correct prediction of the excess precession of Mercury’s orbit around the sun was a first confirmation for the new theory. Another valida- tion was published in 1919, when an expedition lead by Arthur Eddington measured the light deflection by the sun during a total eclipse. At that time astronomers also started to look beyond our local galaxy and realised that the mysterious observed nebulae were nothing more than distant galaxies. Beginning with Edwin Hubble’s observations in the 1920s, first quantitatively relevant measurements of the proper- ties of the Universe became available. After a decade of observations Hubble discovered, in 2 | 1.2. Cosmic Timeline 1929, that distant galaxies are receding from us with a velocity proportional to their distance. Independently, Alexander Friedmann in 1922 derived the so called Friedmann equations from the Einstein field equations, using the metric of a homogeneous and isotropic spacetime and the energy momentum tensor of a perfect fluid. The Friedmann equations in fact could predict the expansion that was observed by Hubble. Another milestone for todays understanding of the evolution of our Universe was the discovery of the cosmic microwave background (CMB) radiation by Penzias & Wilson in 1964. This radiation, which shows an almost perfect Planckian spectrum of TCMB = 2:725 K, is a remnant from the early hot epoch of the Universe when the interstellar gas was still ionised and scattered photons efficiently.
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