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Constraining Extended Theories of Gravity by Large Scale Structure and Cosmography by Vincenzo Salzano

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Constraining Extended Theories of Gravity by Large Scale Structure and Cosmography by Vincenzo Salzano Constraining Extended Theories of Gravity by Large Scale Structure and Cosmography by Vincenzo Salzano Thesis submitted in satisfaction of the ¯nal requirement for the Degree of Philosophi½ Doctor in Fundamental and Applied Physics SUPERVISOR COORDINATOR Salvatore Capozziello Lorenzo Marrucci Universit`adegli Studi di Napoli Federico II Dipartimento di Scienze Fisiche December 2008 Two monks were arguing about a flag. One said: "The flag is moving." The other said: "The wind is moving." The sixth patriarch happened to be passing by. He told them: \Not the wind, not the flag; mind is moving." Mumonkan - Case 29: Not the Wind, Not the Flag \Swans", 1956, wood engraving, Escher M.C. Contents Abstract i List of Publications iii List of Figures v List of Tables ix Part I : General Relativity vs Extended Theories of Gravity 1 1 Introduction 3 1.1 Poetics of Gravity . 3 1.2 Brief History of Gravitational Theory . 4 1.3 Shortcomings of General Relativity . 6 1.4 General Relativity and Quantum Field Theory . 7 1.5 The Standard Cosmological Model . 8 1.5.1 Overview . 8 1.5.2 Problems . 10 1.5.3 Observations . 12 1.6 The ¤CDM model . 16 1.7 Solutions to ¤CDM model? . 18 1.8 Final remarks on General Relativity . 21 2 Extended Theories of Gravity 23 2.1 Dimensional considerations on General Relativity . 23 2.2 Tassonomy and Motivations . 27 2.3 Apparatus . 32 2.4 Extended Theories at cosmological scales . 35 2.4.1 Dark Energy as a curvature e®ect . 35 2.4.2 Reconstructing f(R) model . 42 2.5 Toy models . 45 2.6 Final lesson on Extended theories of Gravity . 50 Part II : Gravitational systems in f(R) 53 3 Galaxies without Dark Matter 55 3.1 Galaxies . 55 3.2 Low energy limit of f(R) gravity . 57 3.3 Extended systems . 61 3.3.1 Spherically symmetric systems . 61 3.3.2 Thin disk . 63 3.4 Low Surface Brightness Spiral Galaxies . 65 3.4.1 Rotation curves: the data . 65 3.4.2 Modelling LSB galaxies . 66 3.4.3 Fitting the rotation curves . 66 3.5 LSB: testing the method . 68 3.5.1 The impact of the parameters degeneracy . 69 3.5.2 Breaking the degeneracy among (¯; log rc; fg) .............. 70 3.5.3 Raw vs smooth data . 72 3.6 Low Surface Brightness Galaxies: Results . 75 3.6.1 Details on ¯t results . 80 3.7 High Surface Brightness Spiral Galaxies . 82 3.8 Burkert haloes . 86 3.9 What have we learnt from galaxies? . 88 4 The Sersic Virial Hyperplane 91 4.1 General properties of Elliptical Galaxies . 91 4.2 Modelling Elliptical Galaxies . 93 4.2.1 The Sersic pro¯le . 93 4.2.2 The dark halo . 94 4.3 The virial theorem . 95 4.3.1 Kinetic and gravitational energy . 96 4.3.2 Scaling relations from the virial theorem . 98 4.3.3 Computing k and w ............................ 100 4.4 Testing the SVH assumptions . 102 4.4.1 The data . 102 4.4.2 Bayesian parameter estimation . 105 4.4.3 Setting the model parameters . 107 4.4.4 Results . 109 4.5 The observed SVH . 112 4.5.1 The SVH coe±cients . 113 4.5.2 A two parameter SVH . 116 4.5.3 The SVH in di®erent ¯lters . 117 4.5.4 The inverse SVH vs the FP and PhP planes . 118 4.5.5 Impact of selection criteria . 121 4.6 Discussion and conclusions . 122 5 Modelling clusters of galaxies by f(R)-gravity 125 5.1 Clusters of Galaxies as fundamental bundle of Dark Matter at large scales . 125 5.2 f(R)-gravity . 126 5.3 Extended systems . 128 5.4 Cluster mass pro¯les . 130 5.5 Galaxy Cluster Sample . 132 5.5.1 Gas Density Model . 132 5.5.2 Temperature Pro¯les . 133 5.5.3 Galaxy Distribution Model . 134 5.5.4 Uncertainties on mass pro¯les . 135 5.5.5 Fitting the mass pro¯les . 136 5.6 Results . 139 5.7 What have we learnt from clusters? . 143 Part III : Cosmography 159 6 Cosmography vs f(R) 161 6.1 Apparatus . 163 6.1.1 Scale factor series . 164 6.1.2 Redshift series . 168 6.2 f(R) derivatives vs Cosmography . 170 6.2.1 f(R) preliminaries . 170 6.2.2 f(R) derivatives with Cosmography . 171 6.3 f(R) derivatives and CPL models . 175 6.3.1 The ¤CDM case . 176 6.3.2 The constant EoS model . 178 6.3.3 The general case . 180 6.4 Constraining f(R) parameters . 181 6.4.1 Double power law Lagrangian . 181 6.4.2 HS model . 183 6.5 Constraints on f(R) derivatives from the data . 185 6.6 Conclusions . 189 7 Conclusions and Perspectives 193 Bibliography 195 Abstract The main aim of this thesis is to show the viability of Extended Theories of Gravity in substi- tuting General Relativity-based cosmological models in the explanation of Universe Dynamics and Origin. After a brief review of all the questions posed by General Relativity and all the possible solutions to these ones, we start to describe deeper one of the alternative approaches to Gravity and Universe, namely the Extended Theories of Gravity. In this context we have chosen to work with a particular class of these theories, the f(R- gravity models, so named because of starting from a general gravitational lagrangian, where the classical R term of the Hilbert-Einstein one is substituted by a general function f(R). We will start with a brief review of history, motivations, pros and cons of these approaches. We also review some results from literature about the cosmological mimicking of dark compo- nents as a curvature (i.e. geometrical) e®ect of f(R) geometry and the possibility to explain rotational curves of spiral galaxies. Then we pass to the original works presented in these pages. We think that we can extract three main characteristics of our work: ² We have never adopted a particular, well de¯ned f(R) model, in contrast with other works; we have always tried to work in the most general hypothesis it was possible. When we will show results of our analysis on clusters of galaxies, it is important to un- derline that they only rely on the assumption of an analytical Taylor expandable f(R), without any other speci¯cation. In the case of cosmological applications, namely in the Cosmography chapter, we could say that we have worked in an even more general scenario: we chose Cosmography because it is a model independent approach to ob- servational data, so we have no basic hypothesis not only on the mathematical form of the f(R)- model, but even on the nature of the universe dynamics (General Relativity or f(R) one); ² Starting from previous point we have tried to give constraints to the hypothetical form of the f(R)-theories: we have derived values of some of the parameters that a viable f(R)-model should have to explain some of the question we have explored (mass pro¯le ii of clusters of galaxies). We have also explored connections between General Relativity- based (dark energy ones) and f(R)-based models (in cosmography) studying their mutual mimicking ability and the possibility of discriminate between them; ² Last, but not least, our analysis has the important merit of having a strong predictive power, so that it can be con¯rmed or confuted by comparing with some tests. From the Cosmography-based analysis, we are strongly dependent on the experimental possibili- ties of future surveys: if we will not have measurements within a certain sensibility range we were not be able (or we will have few possibilities) to discriminate what approach is right and what one is wrong. On the contrary, in the case of clusters of galaxies, we are able to do predicitions on results which could come from the application of f(R)-models to di®erent scales of gravitational systems (galaxies, solar system). If these predictions were exact, then we would have a solid and well founded theoretical model of gravity in alternative to General Relativity. Even if many results we had make us con¯dent to be on the right way, it is important to underline also that our work is always built on a conservative hypothesis. We don't think that f(R) gravity is not the theory of gravity (like General Relativity is not), but it is an important and interesting toy model which can take us nearer an e®ective and deeper comprehension of gravity. List of Publications ² Capozziello, S., Cardone, V.F., Salzano, V., \Cosmographic parameters in f(R) theories", Physical Review D 78, 063504 (2008) ² Capozziello, S., De Filippis, E., Salzano, V., \Modelling clusters of galaxies by f(R)-gravity", Submitted to MNRAS ² Capozziello, S., Cardone, V.F., Molinaro, R., Salzano, V., \The Sersic Virial Hyperplane", Submitted to MNRAS ² Capozziello, S., Lazkoz, R., Salzano, V., \New insights in cosmography with Markov chains", in preparation List of Figures 1.1 History of the Universe. In this schematic key events in the history of universe and their associated time and energy scales are presented. There are also il- lustrated several cosmological probes that provide us with information about the structure and the evolution of the universe. Acronyms: BBN (Big Bang Nuclesynthesis), LSS (Large Scale Structure), BAO.
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