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Cosmological Aspects of Unified Theories

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Cosmological Aspects of Unified Theories University of Patras School of Natural Sciences Department of Physics Division of Theoretical and Mathematical Physics, Astronomy and Astrophysics Cosmological aspects of Unified Theories Andreas Lymperis Supervisor: Prof. Smaragda Lola Thesis submitted to the University of Patras for the Degree of Doctor of Philosophy February 2021 2 Three-member advisory committee (in alphabetical order): 1. Smaragda Lola (Supervisor) - Professor, Physics Department, University of Patras 2. Leandros Perivolaropoulos - Professor, Physics Department, University of Ioannina 3. Andreas Terzis - Professor, Physics Department, University of Patras Seven-member PhD thesis examination committee (in alphabetical order): 1. Spyros Basilakos - Director Of Research, Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens 2. Panagiota Kanti - Professor, Physics Department, University of Ioannina 3. Smaragda Lola (Supervisor) - Professor, Physics Department, University of Patras 4. Leandros Perivolaropoulos - Professor, Physics Department, University of Ioannina 5. Emmanuel Saridakis - Principal Researcher, Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens 6. Vassilis Spanos - Associate Professor, Department of Physics, University of Athens 7. Andreas Terzis - Professor, Physics Department, University of Patras 3 Abstract In this thesis, we study cosmological aspects of some well-known dark energy models, and extending our investigation into Unified Theories. We explore the existence of geodesi- cally complete singularities in quintessence and scalar-tensor quintessence models using analytical expressions for the scale factor and the scalar field. We show that close to the 2 q singularity the scale factor is of the form a(t) = as +b(ts t)+c(ts t) +d(ts t) where − − − as; b; c; d are constants which are obtained from the dynamical equations and ts is the time of the singularity. In the case of quintessence we find q = n + 2 (i.e. 2 < q < 3), while in the case of scalar-tensor quintessence we have a stronger singularity with q = n + 1 (1 < q < 2). We verify these analytical results numerically and extend them to the case where a perfect fluid, with a constant equation of state w = p/ρ, is present. We find that the strength of the singularity (value of q) remains unaffected by the presence of a perfect fluid. The linear and quadratic terms in (ts t) that appear in the expansion of the scale − factor around ts are subdominant for the diverging derivatives close to the singularity, but can play an important role in the estimation of the Hubble parameter. Using the an- alytically derived relations between these terms, we derive relations involving the Hubble parameter close to the singularity, which may be used as observational signatures of such singularities in this class of models. For quintessence with matter fluid, we find that close _ 3 3 2 to the singularity H = Ω0m(1 + zs) 3H . These terms should be taken into account 2 − when searching in cosmological data for such future or past time singularities. Next, using the Tsallis entropy and applying the first law of thermodynamics we construct several cosmological scenarios. We show that the universe exhibits the usual thermal history, with the sequence of matter and dark-energy eras, and according to the value of δ the dark-energy equation-of-state parameter can be quintessence-like, phantom-like, or experience the phantom-divide crossing during evolution. Even in the case where the explicit cosmological constant is absent, the scenario at hand can very efficiently mimic ΛCDM cosmology, and is in excellent agreement with Supernovae type Ia observational data. Finally, we revisit inflation with non-canonical scalar fields by applying deformed steepness exponential potentials. We show that the resulting scenario can lead to infla- tionary observables and in particular to scalar spectral index and tensor-to-scalar ratio in remarkable agreement with observations. Additionally, a significant advantage of the 4 scenario is that parameters, such as the non-canonicality exponent and scale, as well as the potential exponent and scale, do not need to acquire unnatural values and hence can accept a theoretical justification. Hence, we obtain a significant improvement with respect to alternative schemes, and we present distinct correlations between the model parameters that better fit the data, which can be tested in future probes. This combina- tion of observational efficiency and theoretical justification makes the scenario at hand a good candidate for the description of inflation. 5 To my wife Alexandra 6 Whatever tomorrow wants from me At least I'm here, at least I'm free Free to choose to see the signs ....and this is my line. \Poets Of The Fall" 7 Acknowledgements I would like to warmly thank all those that helped, supported, collaborated and believed in me throughout the preparation of this thesis. Without them, it could not have taken its present form. Moreover, during these years, I was given the opportunity to associate with world class scientists and give oral presentations in well-known international conferences, workshops and summer schools. First and foremost, I thank my thesis supervisor Professor S. Lola, for this great opportunity to get involved with physics research. Professor Lola offered me full support, insight into scientific and research subjects and assistance with projects, helping me take my first steps in this unique and beautiful world of physics. She offered me full financial support to travel, meet people and have stimulating and fruitful conversations about several subjects in physics. Above all, she was at my side in difficult times, throughout these years and that makes for more than a typical supervisor - student relationship. Without her assistance and dedicated involvement in every step of the way, this thesis would have never been completed. Even from the first years of my PhD studies I had the luck to meet and collaborate with two outstanding people and scientists, Prof. Leandros Perivolaropoulos and Principal Researcher Emmanuel Saridakis. At the very last, I owe them my gratitude and many thanks for their patience, their trust and for sharing with me their vast knowledge across a broad range of cosmology topics. At this point I would like to also express my gratitude to the other honorable members of my PhD thesis committee, Prof. Andreas Terzis, Prof. Panagiota Kanti, Director of Research Spyros Basilakos and Assistant Prof. Vassilis Spanos. I must also thank my friend and colleague Kostas Blekos, without his help this thesis would not have taken its present form. My words are bound and this acknowledgement would be limited without men- tioning the most basic source of my life, my family. I owe a lot to my wife Alexandra who is my shelter and inspiration all these years and my parents Nickos and Argyroula who encouraged and helped me at every stage of my personal and academic life and longed to see this achievement come true. In addition, I would like to express my gratitude to all the Department faculty mem- 8 bers, for their help and support throughout my years of studying. Lastly, I offer my sincere thanks to all those who directly or indirectly have given me support in this undertaking. Contents 1 Introduction 12 2 The Standard Model of Cosmology 18 2.1 A Brief History Of The Universe . 20 2.2 The ΛCDM Model . 23 2.2.1 The Expanding Universe . 23 2.2.2 Friedmann-Lema^ıtre-Robertson-Walker Metric . 24 2.2.3 Einstein Field Equations . 28 2.2.4 Solutions To Einstein Equations - Friedmann Equations . 29 2.2.5 Problems of the Big-Bang Theory . 33 3 Cosmic Inflation 37 3.1 Canonical Inflationary Dynamics . 38 3.2 Slow-roll Approximation . 40 3.3 Inflationary Observables . 43 3.3.1 Scalar-Tensor Power Spectra . 43 3.3.2 Some Examples Of Canonical Inflationary Models . 46 3.4 Non-canonical Inflationary Dynamics . 49 3.4.1 Non-Canonical Scalar Dynamics . 51 3.4.2 Inflationary Parameters - Observables . 54 3.4.3 Some Examples Of Non-Canonical Inflationary Models . 55 4 Dark Matter 59 4.1 Extensions of the Standard Model . 59 4.2 Evidence For Dark Matter . 60 9 CONTENTS 10 4.2.1 Galaxy Rotation Curves . 60 4.2.2 Gravitational Lensing . 61 4.2.3 CMB Radiation . 62 4.3 Constraints On Dark Matter . 65 4.4 Dark Matter Candidates . 65 5 Dark Energy 68 5.1 Evidence For Dark energy . 68 5.1.1 Luminosity Distance . 68 5.1.2 High-Redshift Supernovae Ia . 70 5.1.3 The Age Of The Universe . 71 5.1.4 CMB and Large-Scale Structure (LSS) . 72 5.2 Dark Energy Candidates . 72 5.2.1 Cosmological Constant . 72 5.2.2 Dark Energy Scalar-Field Models . 74 5.2.3 Quintessence . 74 5.2.4 K-essence . 75 5.2.5 Tachyon field . 75 5.2.6 Phantom field . 76 5.2.7 Chaplygin gas . 76 5.3 The Fate Of The Universe - Future Time Singularities . 77 5.4 Thermodynamical Approach To Dark Energy . 80 5.4.1 Friedmann equations as the first law of thermodynamics . 81 5.5 Modified Gravity . 83 5.5.1 f(R) Gravity . 84 5.5.2 Scalar-Tensor Theories . 85 5.5.3 Gauss-Bonnet Gravity . 86 6 Sudden Future Singularities in Quintessence and Scalar-Tensor Quintessence Models 90 6.1 Sudden Future Singularities in Quintessence Models . 90 6.1.1 Evolution without perfect fluid . 90 CONTENTS 11 6.1.2 Numerical analysis . 99 6.1.3 Evolution with a perfect fluid . 101 6.2 Sudden Future Singularities in Scalar-Tensor Quintessence Models . 105 6.2.1 Evolution without a perfect fluid . 105 6.2.2 Numerical analysis . 108 6.2.3 Evolution with a perfect fluid . 112 7 Modified Cosmology through non-extensive horizon Thermodynamics 117 7.1 Tsallis entropy .
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