Some Aspects in Cosmological Perturbation Theory and f (R) Gravity Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen Friedrich-Wilhelms-Universität Bonn von Leonardo Castañeda C aus Tabio,Cundinamarca,Kolumbien Bonn, 2016 Dieser Forschungsbericht wurde als Dissertation von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Bonn angenommen und ist auf dem Hochschulschriftenserver der ULB Bonn http://hss.ulb.uni-bonn.de/diss_online elektronisch publiziert. 1. Gutachter: Prof. Dr. Peter Schneider 2. Gutachter: Prof. Dr. Cristiano Porciani Tag der Promotion: 31.08.2016 Erscheinungsjahr: 2016 In memoriam: My father Ruperto and my sister Cecilia Abstract General Relativity, the currently accepted theory of gravity, has not been thoroughly tested on very large scales. Therefore, alternative or extended models provide a viable alternative to Einstein’s theory. In this thesis I present the results of my research projects together with the Grupo de Gravitación y Cosmología at Universidad Nacional de Colombia; such projects were motivated by my time at Bonn University. In the first part, we address the topics related with the metric f (R) gravity, including the study of the boundary term for the action in this theory. The Geodesic Deviation Equation (GDE) in metric f (R) gravity is also studied. Finally, the results are applied to the Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime metric and some perspectives on use the of GDE as a cosmological tool are com- mented. The second part discusses a proposal of using second order cosmological perturbation theory to explore the evolution of cosmic magnetic fields. The main result is a dynamo-like cosmological equation for the evolution of the magnetic fields. The couplings between the perturbations in the metric and the magnetic fields are present in the dynamo equation, opening a new perspective in the amplification of magnetic fields at early stages of the universe expansion. The final part of this work is in the field of stellar kinematics in galaxies. It is a project that started at Sternwarte-Bonn Institut some years ago. Here we study the stellar and gas kinematics in HCG 90. Furthermore, we analyze the rotation curves and velocity dispersion profiles for the galaxies in the core of the group. Some possible future applications of the work are discuss. v Contents 1 Introduction 3 2 Field equations and variational principles5 2.1 Introduction........................................5 2.2 Extended theories of gravity...............................6 2.2.1 Cosmological and other motivations.......................7 2.3 Field equations on M ...................................8 2.4 Variational principle in metric f (R) gravity........................ 12 2.4.1 Boundary terms in f (R) gravity......................... 14 αβ σ ασ γ 2.5 Evaluation of the term g (δΓβα) − g (δΓαγ)...................... 15 σ 2.6 Integrals with Mτ and N ................................ 17 2.7 Brans-Dicke gravity.................................... 18 2.7.1 Variation respect to gµν .............................. 18 2.7.2 Variation with respect to φ ............................ 21 2.8 Equivalence between f (R) and scalar-tensor gravity................... 22 2.8.1 Boundary contribution in scalar-tensor gravity.................. 23 2.8.2 Boundary term for metric f (R) gravity...................... 23 2.8.3 Higher order gravities.............................. 24 2.8.4 Some remarks................................... 24 3 Covariant dynamics of the cosmological models: covariant “1+3” formalism 27 3.1 Introduction........................................ 27 3.2 Cosmology in “1+3” language.............................. 27 3.3 The Covariant approach.................................. 28 3.3.1 Space-Time Splitting............................... 29 3.4 The kinematical quantities................................ 30 3.5 Ricci tensor identities................................... 31 3.6 The energy-momentum tensor.............................. 32 3.6.1 The conservation laws.............................. 32 3.7 Einstein field equations in the covariant language.................... 32 3.7.1 The Ehlers-Raychaudhuri equation........................ 33 3.8 The constraint equations................................. 34 3.8.1 Vorticity-free !a = 0 equations......................... 35 3.8.2 Bianchi identities................................. 35 vii 3.9 FLRW cosmologies.................................... 36 3.9.1 Splitting of the FLRW spacetimes........................ 37 3.9.2 The Hubble’s law................................. 38 3.10 Dynamics of FLRW universes.............................. 38 3.11 Cosmological dynamics in metric f (R) gravity...................... 40 3.12 Dynamics in metric f (R) gravity............................. 41 3.12.1 The Gauss equation in f (R) gravity....................... 43 3.12.2 FLRW in metric f (R) gravity........................... 43 3.13 Geodesic deviation equation............................... 45 3.13.1 GDE in the “1+3 formalism”........................... 46 3.14 Geodesic Deviation Equation in f (R) Gravity...................... 48 3.15 GDE in FLRW geometries................................ 49 3.16 Geodesic Deviation Equation in FLRW spacetimes: coordinate method......... 49 3.16.1 Geodesic Deviation Equation for the FLRW universe.............. 51 3.16.2 GDE for fundamental observers......................... 52 3.17 GDE for null vector fields................................. 53 3.17.1 From ν to redshift z ................................ 54 3.17.2 The angular diameter distance DA ........................ 56 3.18 Is it possible a Dyer-Roeder like Equation in f (R) Gravity?............... 57 3.19 An Alternative Derivation................................. 58 3.20 Conclusions and Discussion............................... 58 4 Cosmological Perturbation Theory and Cosmic magnetic fields 61 4.1 Introduction........................................ 61 4.2 Cosmological perturbation theory: linear regime..................... 61 4.2.1 Mathematical background............................ 62 4.3 The perturbed metric tensor................................ 64 4.3.1 Decomposition of perturbations......................... 64 4.3.2 The energy-momentum tensor.......................... 67 4.4 The gauge problem in perturbation theory........................ 67 4.4.1 Gauge transformations and gauge invariant variables.............. 68 4.5 Cosmic magnetic fields.................................. 70 4.6 FLRW background.................................... 72 4.7 Gauge invariant variables at first order.......................... 73 4.7.1 The Ohm law and the energy momentum tensor................. 75 4.7.2 The conservation equations............................ 75 4.8 Maxwell equations and the cosmological dynamo equation............... 76 4.9 Generalization at second order.............................. 77 4.9.1 The Ohm law and the energy momentum tensor: second order......... 78 4.10 The Maxwell equations and the cosmological dynamo at second order......... 80 4.11 Specifying to Poisson gauge............................... 81 4.12 Weakly magnetized FLRW-background......................... 81 4.13 Discussion......................................... 84 4.13.1 Gauge fixing................................... 85 4.13.2 Density evolution................................. 88 viii 5 Kinematics in Hickson Compact Group 90 91 5.1 Introduction........................................ 91 5.2 Galaxy groups....................................... 91 5.2.1 Hickson compact groups............................. 92 5.2.2 HCG90 in the context of HCGs......................... 92 5.2.3 X-ray gas in HCG 90............................... 93 5.2.4 The environment of HCG90........................... 94 5.3 Observations and data reduction............................. 94 5.3.1 Observations................................... 94 5.3.2 Data reduction.................................. 95 5.4 Kinematical analysis................................... 96 5.4.1 Kinematical analysis from MXU data...................... 97 5.5 Long Slit Spectroscopy.................................. 99 5.5.1 Templates..................................... 99 5.5.2 Long-Slit Kinematics.............................. 100 5.5.3 Velocity dispersion profiles............................ 104 5.5.4 Velocity field of the ionized gas......................... 107 5.6 Some remarks....................................... 108 6 Summary & Outlook 109 6.1 Summary Outlook.................................... 109 A Notation and conventions 113 B Basic definitions 117 B.1 Basic definitions...................................... 117 B.1.1 Hypersurfaces.................................. 120 B.1.2 Gauss-Stokes theorem.............................. 122 B.2 A note on the variational principle in field theories.................... 123 B.3 “1+3 definitions”..................................... 125 B.3.1 Useful identities................................. 126 B.3.2 Curvature energy momentum stress tensor.................... 127 B.4 Electrodynamics in the 1 + 3 formalism......................... 127 B.4.1 Maxwell equations................................ 127 B.4.2 Contractions in the GDE: 1 + 3 formalism.................... 128 B.4.3 Contractions in GDE: coordinate method.................... 129 C Some useful results 131 C.1 1 + 3 quantities in CPT.................................. 131 C.1.1 Magnetized fluids................................. 133 Bibliography