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

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 geodesically 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 analytically 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 inflationary 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 combination 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 mentioning 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-Lemaˆıtre-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 ...... 117 7.2 Modified Friedmann equations through the non-extensive first law of thermodynamics ...... 118 7.3 Cosmological evolution ...... 121 7.3.1 Cosmological evolution with Λ =0 ...... 123 6 7.3.2 Cosmological evolution with Λ = 0 ...... 126 7.3.3 Cosmological evolution including radiation ...... 129

8 Inﬂation using non-canonical scalars 136 8.1 Non-canonical inﬂation with deformed-steepness potentials ...... 136 8.2 Results ...... 139

9 Discussion-Conclusions 145 Chapter 1

Introduction

We are at a crossroads of important discoveries and developments in our knowledge of the Universe and how it is evolving. In the last two decades, the discovery of the accelerating expanding Universe, the Higgs particle, gravitational waves, and even the first visual representation of a black hole, led to new theories beyond the standard models of physics and reaffirmed the fundamental pillars of our understanding of the Universe, namely, the Standard Model (SM) of Particle Physics and Einstein’s theory of General Relativity (GR). Despite their successes though, the SM and GR fail to explain some phenomena such as the horizon and flatness problem, the observed matter and the acceleration of the Universe. Recent observational data from the cosmic microwave background (CMB) have revealed that the Universe is almost flat and homogeneous at large scales. The need to solve the above problems has led to new theories as extensions of SM and GR. The theory of cosmic inflation explains the flatness and homogeneity of the Universe in its early moments while it also provides the seeds for the large-scale structure we observe. On the other hand CMB also indicates that around 27% of the total content of the Universe is in a form which does not radiate, dubbed dark matter (DM), while 68% is a mysterious form of energy, dubbed dark energy (DE), which is believed to be the cause of the accelerated expansion of the Universe. Many aspects of DE have been studied so far. There are two approaches that have attracted a lot of attention especially in the past decade. The first approach consists of the so-called finite-time singularities, which are cosmological singularities that happen at

12 CHAPTER 1. INTRODUCTION 13 a finite cosmic time, where geodesics continue beyond the singularity and the Universe may remain in existence. Especially significant is the sudden future singularity because it offers an alternative and smooth fate for the Universe. Also, it is argued that particles crossing the singularity will generate the new geometry of the spacetime, thus providing a “soft rebirth” of the Universe after the singularity crossing. The second approach is the thermodynamical approach of DE. There is a well-known stating that one can express the Einstein equations as the first law of thermodynamics. In the particular case of cosmology in a universe filled with matter and dark energy fluids, one can express the Friedmann equations as the first law of thermodynamics applied on the universe apparent horizon considered as a thermodynamical system. Reversely, one can apply the first law of thermodynamics on the universe horizon, and extract the Friedmann equations. Although this procedure is a conjecture and not a proven theorem, it seems to work perfectly in a variety of modified gravities, as long as one uses the modified entropy relation that corresponds to each specific theory. Inflation and DE are a crucial part of the Standard Model of Cosmology for the reasons we have already mentioned. But several problems have been encountered, including fine-tuning issues large predictions for tensor fluctuations etc. In this respect, theories of scalar fields with non-canonical kinetic terms, as expected in supergravity and superstring theories, including the k-inflation subclass, were found to have significant advantages. These theories arise commonly in the framework of supergravity and string compactifications. Among their many advantages, non-canonical scalars satisfy in a more natural way the slow-roll conditions of inflation, since the additional effective friction terms in the equations of motion of the inflaton slow down the scalar field for potentials which would otherwise be too steep. Hence, the resulting tensor-to-scalar ratio is significantly reduced. Moreover, models with non-canonical kinetic terms often allow for the kinetic term to play the role of dark matter and the potential terms to generate dark energy and inflation In chapter 2 we briefly review the Standard Model of Cosmology, i.e. the ΛCDM model. In chapter 3 we explore the dynamics of cosmic inflation. First, we review the simplest case of a real canonical scalar field minimally coupled to gravity, and then extend the CHAPTER 1. INTRODUCTION 14 analysis to the case of non-canonical scalars, which can drive inflation. In chapter 4 we give an insight into dark matter (DM). DM has been used to describe the weak interaction of a non-relativistic (pressureless) matter with the ordinary (standard) matter. We begin with the history of definition of DM through the motions of galaxies in the Coma Cluster by Fritz Zwicky, continuing with the evidence for DM that comes from the study of the circular velocities of stars and gas i.e. the observations of the rotation curves of galaxies. Furthermore, we review the various constraints on the nature of DM to be consistent with observations closing this chapter with the study of some of the most well studied DM candidates. In chapter 5 we delve into the problem of dark energy (DE). As DM has been used to describe the weak interaction of a non-relativistic (pressureless) matter with the ordinary (standard) matter, the source for the observed late-time acceleration of the Universe was dubbed DE. Next we proceed with the evidence for DE that rely on the changes in the rate of expansion of the Universe and we review some of the most well known DE candidates such as cosmological constant etc. Furthermore, we give a detailed description of exotic cosmological singularities that can determine the fate of the Universe. Next, we give an alternative approach to DE through thermodynamical laws and principals, in which one can express the Einstein equations as the first law of thermodynamics and moreover construct modified gravitational theories without necessarily starting from the standard Einstein-Hilbert action. Closing this chapter we review some of the well known modified gravities and their basic characteristics. In chapter 6 we demonstrate analytically and numerically the existence of geodesically complete singularities in quintessence and scalar tensor quintessence models with a scalar field potential of the form V (φ) φ n with 0 < n < 1. In the case of quintessence, the ∼ | | singularity which occurs at φ = 0, involves divergence of the third time derivative of the scale factor (Generalized Sudden Future Singularity (GSFS)), and of the second derivative of the scalar field. In the case of scalar-tensor quintessence with the same potential and with a linear minimal coupling (F (φ) = 1 λφ), the singularity is stronger and involves − divergence of the second derivative of the scale factor (Sudden Future Singularity (SFS)). In chapter 7 we construct modified cosmological scenarios through the application of the first law of thermodynamics on the universe horizon, but using the generalized, CHAPTER 1. INTRODUCTION 15

nonextensive Tsallis entropy instead of the usual Bekenstein-Hawking one. This yields to modified cosmological equations that contain the usual terms as a particular limit, but which in the general case contain terms that appear for the first time and represent an effective dark energy sector quantified by the nonextensive parameter δ. When the matter sector is dust, we extract analytical expressions for the dark energy density and equation-of-state parameters, and we extend these solutions to the case where radiation is also present. In chapter 8 we revisit inflation with non-canonical scalar fields by applying deformed steepness exponential potentials. A significant advantage of this approach is that free parameters, such as the non-canonicality exponent do not need to acquire unnatural values. The resulting scenario can lead to inflationary observables in remarkable agreement with observations. In chapter 9 we present our conclusions and discuss possible future directions. The Standard Cosmology

Chapter 2

The Standard Model of Cosmology

From the very beginning of our existence as human beings, we have been looking for ways to satisfy our personal thirst for knowledge and understanding of the world we live in. From the simple things we see in front of us to the most complex and exciting about how the universe was born and how is evolving. From the first myths and religions, many years passed until we came to the first scientifically substantiated theories, experiments and observations, thus obtaining a clearer picture of the history of our Universe and its evolution. In particular, the revolutionary theories of the last century still remain the fundamental pillars of modern Physics. In 1915, Albert Einstein published his work entitled General Relativity (GR) [1], which radically changed the image we had until then of our Universe. General Relativity is the geometric description of gravity which more specifically provides a unified description of gravity as a geometric property of space and time, known as spacetime. This geometric description of gravity is provided by elegant field equations known as Einstein’s field equations, which show how matter and energy curve spacetime. The solutions of these equations lead to a Universe that is either contracting or expanding. But this was contrary to Einstein’s idea of a static Universe. In order to accommodate his idea, he introduced in his equations a constant, which later became known as the cosmological constant. At that time, the prevailing view of the cosmos was that the universe consisted entirely of the Milky Way Galaxy. In 1924, Edwin Hubble observed that the majority of galaxies are hurdling away from the Milky Way, which means that the Universe goes beyond the Milky Way galaxy i.e. is expanding. This discovery by Hubble led Einstein to abandon the idea

18 CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 19

of a static universe, saying for the introduction of the cosmological constant that “this was the biggest blunder of my life”. But just before the end of the 20th century, in 1998, another surprising discovery was about to begin a new era in Cosmology and the entire Theoretical Physics. Two independent projects, the Supernova Cosmology Project and the High-Z Supernova Search Team, found that the Universe is not only expanding but it does so at an accelerating rate. This means that the expansion of the Universe is such that the velocity at which a distant galaxy is receding from an observer is continuously increasing with time. Recent observations and data [2] have shown that the Universe is not only made up of the ordinary matter that we have considered so far. On the contrary, it is a very small percentage, around 5%, of the total amount of cosmic energy. The remaining percentage consists of a kind of “matter”, which is called dark matter, and is around 27%, which does not interact with the electromagnetic force. This means it does not absorb, reflect or emit light, making it extremely hard to spot and that’s why it is called dark. The remaining 68% is a mysterious form of energy, called dark energy, which is believed to be a constant energy density that fills the space in a homogeneous way. Dark energy is believed to be the cause of this accelerated expansion of the Universe and has even been linked to the aforementioned cosmological constant. Closing this very short historical introduction, let us imagine that someone has the ability to travel backward in time. What will he see? He will see the planets, galaxies and all the larger extragalactic clusters of astrophysical objects converge in space, as well as spacetime shrink more and more, reaching a unique point known as spacetime singularity (Big-Bang singularity), where density and temperature take indescribably high values. This is the remarkable theory of the Big-Bang. The cosmic microwave background radiation and the cosmological redshift-distance relation are together regarded as the best available evidence for the Big-Bang theory. It was in the mid-1960s when the Cosmic Microwave Background (CMB) [3] provided evidence that the Big-Bang is not a mere theoretical construct, but plays a fundamental role in our efforts to reveal the secrets of our Universe. Measurements of the CMB have made the inflationary Big-Bang theory the so-called Standard Model of Cosmology, also known as the ΛCDM model [4–6] (where Λ refers to the cosmological constant and CDM stands for cold dark matter). CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 20 2.1 A Brief History Of The Universe

As we have already mentioned, the theory of the Big-Bang is so far the most prevalent mechanism that gave birth to our Universe. In this section we will cover a brief history of the Universe from the Big-Bang until the present day. In Fig.(2.1) and Table (2.1) are shown the main stages and basic characteristics of the evolution of the Universe, such as time, temperature and energy.

Era Epoch Time Temperature Energy

Planck t 10−43 s > 1032 K E > 1018 GeV ≤ Grand Uniﬁcation 10−43 t 10−36 s > 1029 K 1016 GeV ≤ ≤ Inﬂationary 10−36 t 10−32 s 1029 K 1016 1014 GeV ≤ ≤ − Electroweak 10−32 t 10−12 s 1028 T 1022 K 100 GeV ≤ ≤ ≤ ≤ Quark 10−12 t 10−6 s > 1012 K 100 MeV ≤ ≤ Hadron 10−6 t 1 s 1012 T 1010 K < 100 MeV ≤ ≤ ≤ ≤ Radiation Era Neutrino decoupling 1 s 1010 K 1 MeV Lepton 1 s t 3 min 1010 T 109 K 1 MeV E 100 KeV ≤ ≤ ≤ ≤ ≤ ≤ Big Bang nucleosynthesis 3 min 3 109 T 5 108 K 100 keV · ≤ ≤ · Matter domination 60 kyr 104 T 3600 K 0.75 eV ≤ ≤ Recombination 260 t 380 kyr 3000 K 0.33 E 0.26 eV ≤ ≤ ≤ ≤ Matter Era Photon decoupling 380 kyr 3000 K 0.28 E 0.23 eV ≤ ≤ Reionization 100 t 400 Myr 19 K 7 E 2.6 meV ≤ ≤ ≤ ≤ Galaxy formation 1 t 9 Gyr 4 K 2.5 E 0.4 meV ≤ ≤ ≤ ≤ Dark energy domination t 9 Gyr < 4 K 0.33 meV ≥ Dark Energy Era Today 13.8 Gyr 2.7 K 0.24 meV

Table 2.1: Epochs of thermal history of the Universe.

Planck epoch: It is the earliest stage of the Big-Bang. It is believed that due to the extraordinary small scale of the Universe at that time, quantum effects of gravity dominated physical interactions. There is no currently available physical theory to describe such short times, and current physics is not predictive. Grand Unification epoch: During this epoch, electromagnetic, strong and weak interactions, are unified. Gravity had separated from the other three fundamental forces at the end of the Planck epoch. Inflationary epoch: During this period the Universe underwent an extremely rapid exponential expansion, in which the scale factor a grew as a(t) = eHt, where t is the cosmic time and H the Hubble parameter. CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 21

Figure 2.1: The evolution of the Universe. Source: Centre for Theoretical Cosmology CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 22

Electroweak epoch: In this epoch the strong force separates from the elecromagnetic and weak force, but electromagnetic and weak interactions remain unified as the electroweak force. Quark epoch: The quark epoch begins approximately 10−12 s after the Big-Bang, and the electroweak force seperates into the electromagnetic and the weak force. In this epoch all the fundamental forces take their present forms. Because of the still high temperature of the Universe, quarks cannot bind together into hadrons. Hadron epoch: In this epoch, the temperature of the Universe has fallen sufficiently for quarks to bind together into hadrons. Neutrino decoupling epoch: At around 1 MeV, neutrinos decouple from the plasma and cease interacting with baryonic matter. Lepton epoch: After the end of the hadron epoch where the majority of hadrons and anti-hadrons annihilated each other, leptons dominate the mass of the universe. Big Bang nucleosynthesis epoch: Around 3 min after the Big Bang [7], the energy of the photons is not enough to break the nuclear binding energy, so that hadron scatterings can now produce the light nuclei, namely deuterium, helium, and lithium. Matter domination epoch: Energy density of matter dominates radiation density and dark energy. Recombination epoch: In this epoch electrons and atomic nuclei first become bound to form neutral atoms. At this point, the photons can no longer break the electron-nucleus binding energy and matter stops being ionized. Photon decoupling epoch: The photons decouple, due to inefficient Thomson scatterings and can now move freely through the Universe. Reionization epoch: In this epoch the matter in the Universe reionizes. Matter perturbations generated during inflation evolve into the first most distant observable objects (stars, galaxies, etc). Galaxy formation epoch: Galaxies coalesce into proto-clusters. Dark energy domination epoch: Dark energy dominates matter and radiation energy density. After this point, the Universe is expanding at an accelerating rate due to the effect of the cosmological constant. Today: Observable Universe. CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 23

Figure 2.2: The anisotropies of the cosmic microwave background, or CMB, as observed by ESA’s (European Space Agency) Planck mission. The CMB is a snapshot of the oldest light in our cosmos, imprinted on the sky when the Universe was just 380000 years old. It shows tiny temperature fluctuations that correspond to regions of slightly different densities, representing the seeds of all future structure: the stars and galaxies of today. This image is based on data from the Planck Legacy release, the mission’s final data release, published in July 2018. Source: ESA and the Planck Collaboration.

After this brief summary of the stages of the Universe during its evolution, we now proceed in the next section to outline the basic ingredients of the Standard Model of Cosmology i.e. the ΛCDM model.

2.2 The ΛCDM Model

2.2.1 The Expanding Universe

The expanding behaviour of the Universe [8, 9], can be parametrized by a dimensionless quantity, known as the scale factor a = a(t). The scale factor can be used to define the so-called comoving distance, which is the distance between two points measured along a path defined at the present cosmological time t0. For an observer at rest, the Universe looks isotropic and homogeneous and it expands in all directions evenly. The comoving coordinates remain fixed, while the physical distance grows as a(t). The two distances CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 24

are related by,

d(t) = a(t)r (2.1) where d is the physical distance and r is the comoving distance. Diﬀerentiating Eq.(2.1) with respect to time we obtain

υ = Hd (2.2) where

a˙ H = (2.3) a is the Hubble parameter and Eq.(2.3) is the famous Hubble Law. Data from recent observations estimate the present value of Hubble parameter to be

H0 = 67.4 km/s/Mpc, (2.4) with a tiny uncertainty of less than 1%. Taking the inverse of the current value of H we can deﬁne the Hubble time

1 tH = = 14.8 billion years. (2.5) H0

The Hubble time is diﬀerent from the age of the Universe t0 13.8 billion years. ≈ Suppose that the universe has been expanding at a constant rate for its entire history. That means a(t) = ct. If we calculate the Hubble time in this model, we get

ct tH = = t 13.8 billion years, (2.6) c ≈ which means that in a linear expansion model, the Hubble time is nothing else but the current age of the Universe.

2.2.2 Friedmann-Lemaˆıtre-Robertson-Walker Metric

One of the fundamental and most important principles in modern Cosmology is the Cosmological Principle. Under this assumption the most important properties of the CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 25

Figure 2.3: Hubble diagram: Velocity-distance relation among galaxies. The black circles and the solid line give the estimation for individual galaxies whereas white circles and the dashed line give the estimation for galaxies combined into groups [10].

Universe are its homogeneity and isotropy. The clearest modern evidence for the Cos- mological Principle comes from measurements of the Cosmic Microwave Background. Homogeneity implies that there are no privileged observers in the Universe (the Universe looks statistically the same from all possible points of view), and isotropy implies there is no preferred direction in which the Universe is observed. We must note that these geometrical properties of the Universe are manifest only at large scales and not at small scales, at which the Universe is inhomogeneous. The geometrical properties of spacetime can be described by a metric, which depends on the spacetime coordinates xµ = (x0, xi), where the ﬁrst component x0 is the time-like coordinate and the second component xi represents the three spatial coordinates. The invariant square of an inﬁnitesimal line element gives the distance between two points and has the form

2 µ ν ds = gµνdx dx (2.7) where gµν is the metric. CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 26

Due to homogeneity and isotropy, the metric gµν must be a 4 4 tensor with 10 × independent coeﬃcients, 4 diagonal and 6 oﬀ-diagonal. The metric gµν which describes a homogeneous and isotropic Universe is called Friedmann - Lemaˆıtre - Robertson - Walker (FLRW) metric [11–17]. From the property of isotropy, since there are no preferred

directions, the oﬀ-diagonal components of gµν (µ = ν) must vanish and furthermore from 6 the property of homogeneity, since there are no privileged observers in the Universe, the

metric gµν must not depend on the spatial coordinates. Taking the local limit in the Minkowski spacetime, GR can be approximated by the special theory of relativity, and for a local observer the FLRW metric can be approximated by the Minkowski metric

ηµν = diag( 1, +1, +1, +1) as −   1 0 0 0 −   2   0 a (t) 0 0  gµν =   (2.8)  2   0 0 a (t) 0    0 0 0 a2(t) where a(t) is the scale factor and eq.(2.7) becomes

2 2 2 i j ds = dt + a (t)δijdx dx (2.9) −

where δij = diag(+1, +1, +1) is the Kronecker delta in an Euclidean space. To describe the spatial curvature, we introduce a new parameter k = 1, 0, +1, where − k = 1 refers to an open, k = 0 refers to a ﬂat and k = +1 refers to a closed Universe − respectively. Using standard spherical coordinates (r, θ, φ), the line element (2.9) becomes

dr2 ds2 = dt2 + a2(t) + r2(dθ2 + sin2 θdφ2) . (2.10) − 1 kr2 − Given the metric gµν it is possible to study the free motion of a particle in the spacetime. In GR, a freely moving or falling particle always moves along a geodesic, which is the equation of motion of freely-falling particles in curved spacetime. Particles travelling on a geodesic are free from all external, non-gravitational forces. The geodesic equation can be written as

d2xµ dxα dxβ + Γµ = 0 (2.11) dλ2 αβ dλ dλ CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 27

ρ where λ is a scalar parameter of motion and Γµν are the Christoﬀel symbols which are symmetric in the µ and ν indices and deﬁned as:

gρτ Γρ = (∂ g + ∂ g + ∂ g ) (2.12) µν 2 µ ντ ν µτ τ µν µ where we introduced the notation ∂µgντ = ∂gντ /∂x . Using the metric gµν we can compute the geodesics by calculating the components of the Christoﬀel symbols. In FLRW spacetime, the components of the Christoﬀel symbols are calculated to be:

0 0 Γ0µ = Γµ0 = 0 , (2.13)

0 Γij = δijaa˙ , (2.14) a˙ Γi = Γi = δ , (2.15) 0j j0 ij a i Γαβ = 0 otherwise. (2.16)

The Christoffel symbols are also important and necessary to define another crucial quantity, the Ricci tensor, which in turn is very important for the Einstein equations. The Ricci tensor is defined by the following expression:

α α α β α β Rµν = ∂αΓ ∂νΓ + Γ Γ Γ Γ (2.17) µν − µα βα µν − βν µα and like the Christoﬀel symbols, is also symmetric in the µ and ν indices. Taking the trace of the Ricci tensor we obtain the Ricci scalar:

µ µν R = R µ = g Rµν, (2.18)

µν where g is the inverse of gµν. In FLRW spacetime the Ricci tensor is diagonal and its components are

a¨ R00 = 3 , (2.19) − a 2 Rij = δij aa¨ + 2˙a + 2k , (2.20) while the Ricci scalar is CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 28

a¨ a˙ 2 k R = 6 + + . (2.21) a a2 a2 With these quantities at hand, we can now proceed to the next chapter and obtain the ﬁeld equations of GR, i.e. Einstein’s ﬁeld equations.

2.2.3 Einstein Field Equations

The Einstein ﬁeld equations relate the geometry of spacetime with the distribution of matter within it. In order to obtain the ﬁeld equations, we start from the full action of

the theory, which is the sum of the Einstein-Hilbert action plus a term m describing any L matter ﬁelds appearing in the theory, which reads

Z 4 1 = d x√ g (R 2Λ) + m , S − 2κ − L Z Z 1 4 4 = d x√ g(R 2Λ) + d x√ g m = EH + m , (2.22) 2κ − − − L S S

1 where κ = 8πG = 2 is Einstein’s constant, g = det(gµν) is the determinant of the Mpl metric tensor matrix, R is the Ricci scalar and Λ is the cosmological constant. Varying the action (2.22) with respect to the metric gµν and using the action principle we obtain Einstein’s ﬁeld equations

Gµν + Λgµν = 8πGTµν (2.23)

where Gµν is the Einstein tensor, which describes the spacetime curvature in a manner consistent with energy and momentum conservation, is symmetric in the µ and ν indices,

µν divergenceless ( µG = 0) and is deﬁned as ∇

1 Gµν = Rµν gµνR. (2.24) − 2 Substituting the expression of Einstein’s tensor in eq.(2.23) we have

1 Rµν gµνR + Λgµν = 8πGTµν. (2.25) − 2

Tµν is the energy-momentum tensor, which describes the density and ﬂux of energy and momentum in spacetime and is symmetric in the µ and ν indices. Assume that the CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 29

Universe is ﬁlled with a perfect, isotropic and homogeneous ﬂuid. Then the energy- momentum tensor can be expressed as

Tµν = (ρm + p)UµUν + pgµν (2.26)

where ρ is the energy density and p the pressure of the ﬂuid respectively and Uµ is

the 4-velocity of the perfect ﬂuid, which is relative to the observer. Choosing Uµ =

(1, 0, 0, 0) for an observer comoving with the ﬂuid, and the signature of the metric gµν to be ( 1, +1, +1, +1), the energy-momentum tensor can be written in the form −

Tµν = diag(ρ, p, p, p) (2.27)

and in a matrix form

  ρ 0 0 0     0 p 0 0 Tµν =   . (2.28)   0 0 p 0   0 0 0 p Closing this section, let us mention that in the absence of any matter sources i.e.

Tµν = 0, the Einstein ﬁeld equations (2.25) become

1 Rµν gµνR + Λgµν = 0. (2.29) − 2

2.2.4 Solutions To Einstein Equations - Friedmann Equations

Let us write down the time-like and spatial-like component of the Einstein tensor (2.24). We obtain

2 k ˙ 2 k G00 = 3 H + ,Gij = gij 2H + 3H + . (2.30) a2 − a2 Combining eq.(2.30) with eq.(2.28), for the 00 and the ii components, we obtain two diﬀerent independent diﬀerential equations, the so-called Friedmann Equations:

a˙ 2 8πG k Λ H2 = ρ + (2.31) ≡ a 3 − a2 3 CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 30

a¨ 4πG Λ H˙ + H2 = (ρ + 3p) + (2.32) ≡ a − 3 3 P P where ρ = i ρi and p = i pi are the total energy density and pressure of the Universe respectively, which are the contributions from various matter and energy species (dark energy, baryons, dark matter, photons, neutrinos, etc.). Now by combining Eq. (2.31) with Eq.(2.32) we obtain the so-called continuity equation:

ρ˙ + 3H(ρ + p) = 0. (2.33)

Let us mention, that the continuity equation can be also derived by the covariant conservation of the energy-momentum tensor, i.e.

µ µ µ α α µ DµT ∂µT + Γ T Γ T = 0. (2.34) ν ≡ ν αµ ν − µµ α Eqs.(2.31), (2.32) and (2.33) are the cosmological equations that describe the evolution of the Universe. Of course as the continuity equation (2.33) can be derived from Eqs.(2.31) and (2.32), two of Eqs.(2.31), (2.32) and (2.33) are independent. Nevertheless, it is quite useful to determine a relationship between energy density ρ and the scale factor a(t). It proves useful to deﬁne a new parameter w, the so-called equation of state parameter, which in FLRW cosmology describes the evolution of the Universe. The equation of state parameter of a barotropic perfect ﬂuid has the following simple expression:

p w = , (2.35) ρ where w is assumed to be constant. For a ﬂat universe (k = 0), inserting Eq.(2.35) into Eq.(2.33) we have

a˙ ρ˙ + 3 (1 + w)ρ = 0 (2.36) a where we have used the fact that H =a/a ˙ . Neglecting the cosmological constant terms, and solving the Friedmann equations (2.31) and (2.32) we obtain

ρ a−3(1+w), (2.37) ∝ and CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 31

2 a(t) t 3(1+w) . (2.38) ∝ The above solution is valid for w = 1. Moreover, in a radiation-dominated Universe 6 − w = 1/3, whereas in a dust-dominated Universe w = 0, and in these cases the above solutions for ρ and a(t) can be written as:

−4 1 Radiation : ρ a , a(t) t 2 , (2.39) ∝ ∝ −3 2 Dust : ρ a , a(t) t 3 . (2.40) ∝ ∝

Both cases above, where w 0, correspond to a decelerated expansion of the Universe. ≥ As we have already mentioned, the Universe is undergoing an accelerated expansion, according to the latest observations. In this case, wherea ¨ > 0, an accelerated expansion occurs for

1 w < , (2.41) −3 in which condition (2.41) we require a large negative pressure in order to give rise to an accelerated expansion. The more exotic case w = 1 correspons to a constant energy density ρ, as we can − see from Eq.(2.36). This case leads to an exponential expansion as in the case of a cosmological constant-dominated Universe which is the de-Sitter Universe. Its solution can be written as

0 Ht ρΛ a , a(t) exp (2.42) ∝ ∝ The component that substitutes the cosmological constant is usually referred to as Dark Energy (DE). We will return to this subject in Chapter 5. Some other cases are the contribution of the curvature fluid with w = 1/3 and the case with w < 1, which − − corresponds to a phantom (ghost) dark energy. In this context, of a flat and zero cosmological constant Universe, from Eq.(2.31), one can define the critical density of the Universe, which is the mean density of matter that is required for gravity to halt the expansion of the Universe, to be CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 32

2 3H today 2 ρc = = ρ0c = 3H , (2.43) 8πG ⇒ 0

where H0 is the present value of the Hubble parameter and the value of critical density −29 3 is about ρc 10 g/cm . If the cosmological constant is zero, a universe with a ≈ density below the critical density will expand for ever, whereas a universe with a density greater than the critical density will eventually collapse. A universe with exactly the critical density is described by the Einstein de-Sitter model, which lies on the dividing line between these two extremes. The average density of material that can be directly observed in our Universe is less than 1% of the critical value. However, various observations suggest that most of the matter in the Universe is dark matter, which cannot be seen directly but which can be detected through its gravitational eﬀect. Even when the dark matter is added in, however, the average density of the Universe is still only about 30% of the critical density. It therefore seems likely that the Universe will expand for ever, especially because recent observations imply that the cosmological constant is not zero, which will cause the expansion to accelerate.

Taking the ratio between the energy density of the various species ρi and the critical

density ρc, we can deﬁne the density parameter

ρi Ωi = (2.44) ρc where the subscript i corresponds to the various contributions for example, i = r for radiation, i = k for curvature, i = m for matter, i = Λ for cosmological constant, etc. In the absence of a cosmological constant, a Universe with Ω < 1 will expand for ever, whereas a Universe with Ω > 1 will eventually collapse. Current observations imply that the matter density parameter (baryonic and cold dark matter) is about Ω0m ' Ω0CDM + Ω0b 0.3. The Universe will therefore expand for ever. The observations also ≈ imply that the Universe contains something like a cosmological constant, which is causing

this expansion to accelerate and its value is about Ω0Λ 0.7. A Universe with Ω0tot = 1 ≈ is ﬂat and contains enough matter to halt the expansion but not enough to recollapse it. Inserting the density parameter of the various species into the Friedmann equation (2.31) we obtain CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 33

2 H −2 −3 −4 2 = Ω0Λ + Ω0ka + Ω0ma + Ω0ra . (2.45) H0

Furthermore, if we use the normalized value for today of the scale factor (a0 = a(t0) = 1), we obtain the following relation between all the density parameters

Ω0tot = Ω0Λ + Ω0k + Ω0m + Ω0r = 1. (2.46)

Current cosmological observations show that we live in a ﬂat, Ω0tot = 1 Universe. Lastly, in FLRW cosmology, it proves convenient to introduce the decelaration parameter, which is a dimensionless measure of the cosmic acceleration of the expansion of the Universe, and can be deﬁned as

H˙ q 1 , (2.47) ≡ − − H2 and in terms of Ω and w parameters

1 X q Ω (1 + 3w ). (2.48) 2 i i ≡ i The deceleration parameter is positive for a decelerated expansion and negative for an accelerated expansion of the Universe.

2.2.5 Problems of the Big-Bang Theory

Despite of the remarkable successes of the Big-Bang theory, three significant problems emerged, which are mostly connected to the initial conditions of the Universe. This was the original motivation for the remarkable theory of Inflation, which proposes that the Universe underwent a period of extremely rapid (exponential) expansion, during its first few moments.

The Flatness Problem

Recent observations have determined the geometry of the universe to be nearly flat. However, it is known that curvature grows with time under Big-Bang theory. It would require an extreme fine-tuning of initial conditions to yield a Universe as flat as we see it today, which would be an unbelievable coincidence. CHAPTER 2. THE STANDARD MODEL OF COSMOLOGY 34

The Horizon Problem

The uniformity of the CMB temperature, Fig.(2.2), shows that distant regions in the Universe, which are in opposite directions of the sky and far apart from each other, must have been in contact with each other in the past. According to standard Big Bang expansion theory, these regions could never have been in causal contact with each other. This is because the time needed for light to travel between them exceeds the age of the Universe.

The Monopole Problem

In the very early stages of the Universe, Big Bang cosmology predicts, the production of some heavy, stable exotic particles known as magnetic monopoles. However, magnetic monopoles have never been observed, and this fact has established them as one of the significant problems of the Big-Bang theory. Inflation naturally explains the above problems and furthermore explains the origin of structure in the universe and links important ideas in modern physics, such as symmetry breaking and phase transitions to Cosmology. In the next chapter we will give an insight into this remarkable theory of Inflation, which is considered an extension of the Big-Bang Cosmology. Beyond The Standard Cosmology

Chapter 3

Cosmic Inﬂation

As already stated, the revolutionary idea of Inflation provides not only a solution to the flatness and horizon problem [18, 19], but also it provides the mechanism that produces the primordial seeds of the large structure we observe today in the Universe [20–25]. Independently of the nature of the mechanism, it consists of a short accelerating epoch soon after the Big-Bang, during which the Universe exponentially increased, i.e. inflated. In the inflationary stage the scale factor grows exponentially giving rise to the accelerated expansion in the early Universe. The necessary condition for Inflation, follows from the fact that the change of the shrinking Hubble radius (aH)−1 over time is

d 1 a¨ = < 0, (3.1) dt aH −(aH)2 and therefore, from the inequality we obtain

a¨ > 0, (3.2) which means that accelerated expansion is a requirement for the solution of the flatness and horizon problem. The accelerated expansion then requires a source with an equation of state w < 1 − 3 as stated in 2.2.4. But the question that arises is which ingredient can be responsible § for the inflation of the early Universe? The simplest component of a cosmic fluid that possesses large negative pressure, minimally coupled to gravity, is a scalar field φ the so- called inflaton. The scalar field’s potential dominates over its kinetic energy for a short

37 CHAPTER 3. COSMIC INFLATION 38

period of time and drives the accelerated expansion of the Universe. In this context, the inﬂaton is referred to as the simplest candidate for studying the dynamics of inﬂation.

3.1 Canonical Inﬂationary Dynamics

At the backround level, we consider a homogeneous scalar ﬁeld φ(t, xi) = φ(t), whose Lagrangian density is given by

1 µν = g ∂µφ∂νφ V (φ) (3.3) L 2 − where the first term on the right hand side, is the kinetic energy of the scalar field and V (φ) is the potential of the inflaton. By minimally coupling the inflaton with gravity, we obtain the following action

Z 4 1 1 µν = d x√ g R + g φ;µφ;ν V (φ) = EH + φ, (3.4) S − 2κ 2 − S S where we have assumed zero cosmological constant and no other matter sources other than that of the single inﬂaton’s.

Varying the action (3.4) with respect to the metric gµν we obtain the usual Einstein tensor 1 Rµν gµνR as we showed in 2.2.3, plus a term that deﬁnes the energy-momentum tensor − 2 § of the scalar ﬁeld, which reads as

φ 2 δ φ 1 λ Tµν = S = ∂µφ∂νφ gµν ∂ φ∂λφ + V (φ) . (3.5) −√ g δgµν − 2 − On the other hand, by varying Eq.(3.4) with respect to the scalar ﬁeld φ we obtain

δ φ 1 µ dV (φ) S = ∂µ(√ g∂ φ) + = 0. (3.6) δφ √ g − dφ − In a FLRW Universe, described by the metric (2.10), the 00 and ii components of Eq.(3.5), can be related to those in Eq.(2.26) for a perfect ﬂuid and give

1 ρ = T 0 = φ˙2 + V (φ) , (3.7) φ 0 2 i 1 ˙2 pφ = T = φ V (φ) , (3.8) − i 2 − CHAPTER 3. COSMIC INFLATION 39

while from Eq.(3.6) we obtain the Klein-Gordon equation for a scalar ﬁeld in the gravitational background

φ¨ + 3Hφ˙ + V 0 = 0, (3.9) where prime denotes differentiation with respect to the scalar field φ. Note that the Klein-Gordon equation (3.9) can be also obtained through the continuity equation (2.33) by substituting Eqs.(3.7), (3.8). Furthermore, the Friedmann equations (2.31), (2.32), using the equations (3.7), (3.8) and assuming a flat Universe take the form

8πG 1 H2 = φ˙2 + V (φ) (3.10) 3 2 and

H˙ = 4πGφ˙2. (3.11) − Eqs.(3.10), (3.11), together with Eq.(3.9), determine the dynamics of the scalar ﬁeld in the gravitational background and hence are the so-called background gravitational equations of motion. Lastly, the equation of state parameter for the scalar ﬁeld takes the following form

1 ˙2 pφ 2 φ V (φ) wφ = = − . (3.12) ρ 1 ˙2 φ 2 φ + V (φ) It is more than obvious, that the inﬂaton can produce the necessary negative pressure i.e.

1 wφ < 0, and lead to an accelerated expansion i.e. wφ < , if its potential dominates − 3 over its kinetic energy for this short period of time (inflationary era). To summarize, and as we will show in the next section, whatever the mechanism for inflation is, in this context the following conditions for inflation must be simultaneously satisfied:

H˙ 1 a¨ > 0, H = < 1, wφ < . (3.13) −H2 −3 CHAPTER 3. COSMIC INFLATION 40

Figure 3.1: The standard picture of inflation. The inflaton rolls down its potential, inflating the Universe. A sufficiently flat plateau would make the field roll slowly towards

the minimum. After inﬂation ends when H = 1, the inﬂaton oscillates around the potential’s minimum and fuels the reheating era of the Universe.

3.2 Slow-roll Approximation

In the inflationary paradigm, there are two vacuum states, i.e. false and true vacuum. Inflation starts when the scalar field begins to move out of the false vacuum, slowly rolling down to the true vacuum, subject to the slow-roll approximation Fig. 3.1 [26–28]. Let us recall the second Friedmann equation (2.32) and rewrite it as

a¨ 2 = H (1 H ), (3.14) a − where we have deﬁned [29]

H˙ 3φ˙2 H = = . (3.15) −H2 φ˙2 + 2V Eq.(3.15) is the so-called ﬁrst Hubble slow-roll parameter. In addition, the third condition from Eqs.(3.14) can be recast as φ˙2 < V (φ). Inserting this constraint into the ﬁrst Hubble slow-roll parameter i.e. into Eq.(3.15), the second condition of Eqs.(3.14) can be recast CHAPTER 3. COSMIC INFLATION 41

as the ﬁrst slow-roll condition obtaining

3φ˙2 H = < 1 (3.16) φ˙2 + 2V

while inﬂation ends exactly when H = 1. Furthermore, in the limit H 0, the potential → energy of the scalar ﬁeld dominates its kinetic energy, implying the even stronger condition

φ˙2 V (φ). (3.17) Moreover, to have a successful inﬂation, the acceleration of the inﬂaton i.e. φ¨, must be negligible compared to the rate of the expansion, and can be expressed as

φ¨ 3Hφ˙ , V 0(φ) . (3.18) | | | | | | Using the above inequality, we can introduce the second Hubble slow-roll parameter

H¨ φ¨ ηH = = , (3.19) −2HH˙ −Hφ˙ where now the second slow-roll condition for ηH reads as

ηH < 1. (3.20) | | The requirement that the potential energy V (φ) of the inflaton should dominate its kinetic energy φ˙2/2, imposes the requirement that the scalar field should roll slowly down its potential. This means that the the slow-roll conditions Eqs.(3.16),(3.20) place constraints on the velocity of the inflaton. This can been seen in Fig. 3.1, where a sufficiently flat potential would make the field roll slowly towards the minimum i.e. true

vacuum. After inﬂation ending at H = 1 the ﬁeld starts to oscillate around the minimum and the Universe starts to reheat [30–32]. Within the slow-roll conditions (3.17) and (3.18), the Friedmann (3.10) and Klein- Gordon (3.9) equations can be written as

8πG H2 V (φ) , (3.21) ≈ 3 V 0 φ˙ , (3.22) ≈ −3H CHAPTER 3. COSMIC INFLATION 42

respectively. Eqs.(3.21), (3.22) constitute the so-called slow-roll approximation. From Eq.(3.21) one can easily see that the Hubble parameter H is approximately constant during the inﬂationary era as expected. Moreover, Eqs.(3.21), (3.22) allow us to deﬁne the Hubble slow-roll parameters as parameters of the shape of the potential V (φ), also known as potential slow-roll parameters, which in this case can be written as

M 2 1 ∂V 2 M 2 ∂2V = pl , η = pl . (3.23) V 2 V ∂φ V V ∂φ2 In the slow-roll regime i.e. as long as the slow-roll approximation holds, the potential slow-roll parameters can be related to the Hubble slow-roll parameters as

H V , (3.24) ≈

ηH ηV V , (3.25) ≈ − which are also subject to the slow-roll conditions i.e. inflation ends when V , ηV 1. ≈ The above relations are usually assumed to be sufficient for the determination of the inflationary observables. Lastly, during inflation the exponential change of the scale factor can be measured and parametrized by a dimensionless number, the so-called number of e-folds, which is defined as

Z tend N ln(aend/aini) = Hdt, (3.26) ≡ tini where aini, aend is the scale factor at the beginning and at the end of inﬂation respectively, and tini, tend is the time of the beginning and the end of inﬂation respectively. In the slow-roll approximation, the number of e-folds can be written as

Z tend Z φend H Z φ V N = Hdt = dφ dφ, (3.27) ˙ 0 t φ φ ≈ φend V where we have also used the derivative chain rule dt = (dt/dφ)dφ = dφ/φ˙. In terms of the slow-roll parameters the e-folding number becomes

Z φ dφ Z φ dφ N = . (3.28) φend √2H ≈ φend √2V CHAPTER 3. COSMIC INFLATION 43

The value of e-foldings needed for the horizon and ﬂatness problem to be solved is calculated to be around N 50 60 e-folds [33,34], which in turn is also consistent with the ≈ − generation of the CMB anisotropies.

3.3 Inﬂationary Observables

In every inﬂationary scenario the important quantities are the inﬂation-related observables, namely the scalar spectral index of the curvature perturbations nS and its running

αS, the tensor spectral index nT and its running αT , as well as the tensor-to-scalar ratio r. In order to extract the relations for the inflation-related observables, a detailed and thorough perturbation analysis is needed. In the simple case of canonical fields minimally coupled to gravity, and introducing the slow-roll parameters, full perturbation analysis indicates that the inflationary observables can be expressed solely in terms of the scalar potential and its derivatives. Since we will not develop the full pertubation analysis and calculations, we suggest further readings for more details, e.g. [26, 28, 35] and the references therein.

3.3.1 Scalar-Tensor Power Spectra

Using scalar-vector-tensor decomposition, in a spatially ﬂat FLRW Universe, the perturbed line element has the form [36–39]

2 2 i 2 i j ds = (1 + 2A)dt 2a(t)(∂iB)dtdx a (t)[(1 2ψ)δij + 2(∂i∂jE) + hij]dx dx (3.29) − − − where A, B, ψ, and E are the scalar degrees of freedom, while hij are the tensor degrees of freedom. We consider only scalar and tensor pertubations since it is well known that vector perturbations are not produced during scalar field inflation. The curvature perturbation , which geometrically measures the spatial curvature of comoving hyper- R surfaces (surfaces of constant φ), is defined as a gauge invariant combination of the metric perturbation ψ and the scalar field perturbation δφ, namely

H ψ + δφ (3.30) R ≡ φ˙ CHAPTER 3. COSMIC INFLATION 44

Using Eq.(3.30), the perturbed linearized Einstein’s equation δGµν = κδTµν and the equation that governs the evolution of perturbations in the scalar ﬁeld we obtain [37]

0 00 z 0 2 2 + 2 + c k k = 0 (3.31) Rk z Rk s R R dt where prime denotes derivative with respect to conformal time τ = a(t) , k is the measure ~ of the wave number k, cs is the eﬀective speed of sound of the scalar ﬁeld perturbation, which is given by [40]

(∂ /∂X) c2 = L (3.32) s (∂ /∂X) + (2X)(∂2 /∂X2) L L and z is given by

a(ρ + p )1/2 z = φ φ . (3.33) csH

Let us note here that for canonical scalar models e.g. quintessence or phantom etc, cs = 1.

In terms of the Mukhanov-Sasaki variable uk = z k [36, 37, 41], Eq.(3.31) can be R rewritten as

00 00 2 2 z u + c k uk = 0 (3.34) k s − z and the corresponding equation governing the tensor perturbations has the form

00 00 2 a υ + k uk = 0, (3.35) k − a h where h is the amplitude of the tensor perturbation and υk = a . The power spectrum of scalar curvature perturbations is deﬁned as

3 3 2 k 2 k uk S(k) = k = | | (3.36) P 2π2 |R | 2π2 z while the power spectrum of tensor perturbations is deﬁned as

3 3 2 k 2 k υk T (k) = 2 hk = 2 | | . (3.37) P 2π2 | | 2π2 a The expressions for scalar and tensor power spectrum in the slow roll limit can be written as CHAPTER 3. COSMIC INFLATION 45

H2 2 (k) = (3.38) S 2π[c (ρ + p )]1/2 P s φ φ aH=csk and

! ! 8 H 2 2V (φ) T (k) = (3.39) P M 2 2π ' 3π2M 4 pl aH=k pl aH=k respectively. We have to note that in the canonical case of a single scalar field inflation the speed of sound (3.32) is equal to unity i.e. cs = 1. Furthermore, the deviation of the scalar power spectrum from scale invariance is measured by the scalar spectral index nS, which is defined as

d ln S nS = 1 + P = 1 4H + 2ηH = 1 6V + 2ηV (3.40) d ln k − − where exact scale invariance corresponds to nS = 1. Its running is deﬁned as

dn α = S . (3.41) S d ln k The deviation of the tensor power spectrum from scale invariance is measured by the

tensor spectral index nT , which is deﬁned as

d ln T nT = P = 2H = 2V (3.42) d ln k − −

where exact scale invariance corresponds to nT = 0 and its running is deﬁned as

dn α = T (3.43) T d ln k Lastly, the tensor-to-scalar ratio is deﬁned as

T r = P = 16H = 16V (3.44) S P where in the slow roll regime we can obtain the following consistency relation

r = 8nT . (3.45) − In a given scenario these quantities depend on the model parameters, and hence confrontation with observational data can lead to constraints on these model parameters. CHAPTER 3. COSMIC INFLATION 46

3.3.2 Some Examples Of Canonical Inﬂationary Models

Chaotic Inﬂation

Chaotic inﬂation is usually associated with power law potentials, which are of the following form

n V (φ) = V0φ , where V0, n > 0. (3.46)

In order to calculate the scalar spectral index in the context of the chaotic inflation, we shall first obtain an expression for the scalar field in terms of the number of inflationary e-folds to the end of inflation i.e. φ(N). As previously mentioned, inflation ends when

slow-roll parameters V , ηV grow approaching the value of unity. In this case the slow-roll parameters (3.23) take the following form

n2 1 = , V 2 φ2 1 ηV = n(n 1) . (3.47) − φ2

Substituting V = 1 in the above equation, we obtain the following expression for the value of the scalar ﬁeld when inﬂation ends

φ n e = . (3.48) Mpl √2 In the context of the slow-roll approximation and by using the relation of the number of the e-folds (3.27) we obtain the following simple expression

1 φ h ni 2 N = 2n N + . (3.49) Mpl 4 Furthermore, inserting the expression of the potential (3.46) into Eq.(3.38) of the scalar power spectrum we have

! 1 6 V0 S(k) = 2 2 4−n . (3.50) P 72π cs n Mpl Then using Eq.(3.40) we ﬁnd for the scalar spectral index CHAPTER 3. COSMIC INFLATION 47

2n + 4 nS = 1 (3.51) − 4N + n where we have used the relation

d d . (3.52) d ln k ' −dN

Moreover, the running of the spectral index αS given by Eq.(3.41) reads as

2 2 αS = (nS 1) . (3.53) −2 + n − Turning now to the tensor power index (3.40) we get

2n nT = , (3.54) −4N + n while for the tensor-to-scalar ratio (3.44) we ﬁnd

16n r = . (3.55) 4N + n Let us now consider N = 60 and substitute n = 2 in Eq.(3.46) which corresponds to

1 2 2 quadratic potential V (φ) = 2 m φ and n = 4 which corresponds to quartic potential 1 4 V (φ) = 4 λφ respectively. For the scalar spectral indexes and tensor-to-scalar ratios, in these cases, we obtain

Quadratic potential V (φ) = 1 m2φ2: • 2

4 nS = 1 (3.56) − 2N + 1

and

16 r = . (3.57) 2N + 1

For N = 60 we ﬁnd

nS 0.97, r 0.13. (3.58) ' ' CHAPTER 3. COSMIC INFLATION 48 20 .

0 Convex TT,TE,EE+lowE+lensing TT,TE,EE+lowE+lensing +BK15 Concave TT,TE,EE+lowE+lensing +BK15+BAO 15 . )

0 Natural inﬂation 002 .

0 Hilltop quartic model r α attractors Power-law inﬂation 2 10 R inﬂation . 0 V φ2 ∝ V φ4/3 ∝ V φ ∝ 2/3 Tensor-to-scalar ratio ( V φ 05 . ∝ 0 Low scale SB SUSY N =50 ∗ N =60 ∗ 00 . 0 0.94 0.96 0.98 1.00

Primordial tilt (ns)

−1 Figure 3.2: Marginalized joint 68% and 95% CL regions for nS and r at k = 0.002 Mpc from Planck 2018 [42] compared to the theoretical predictions of selected inﬂationary models. Source: [42]

Quartic potential V (φ) = 1 λφ4: • 4

3 nS = 1 (3.59) − N + 1

and

16 r = . (3.60) N + 1

For N = 60 we ﬁnd

nS 0.95, r 0.26. (3.61) ' '

Comparing these values with the latest results from the Planck 2018 [42] (Fig.3.2), we can easily see that both models are essentially ruled out, since they predict a tensor-to-scalar ratio which is too large. As we will see in the next section, modifying the scalar ﬁeld’s CHAPTER 3. COSMIC INFLATION 49 kinetic term namely non-canonical scalars, improves the above results for the inﬂationary observables, in agreement with observations.

The Exponential Potential

Large ﬁeld inﬂation can be also associated with another type of potential, the so-called exponential potential, which has the following form:

r2 φ V (φ) = V0 exp (3.62) − q Mpl Due to power law (a(t) tq) expansion of a spatially ﬂat Universe, dominated by a ∝ single scalar ﬁeld, we can see that the slow-roll parameters (3.15), (3.19) and (3.23) are

1 constants i.e. H = V = q . This means that there is no successful exit from inﬂation. But as we will see in the next section this is not the case for non-canonical models.

3.4 Non-canonical Inﬂationary Dynamics

As already mentioned, inflation is a crucial part of the Standard Model of Cosmol- ogy [18, 19, 43–45]. Its solution to the horizon and flatness problems, together with the predictions for an almost scale invariant perturbation spectral index, have been confirmed by measurements of the cosmic microwave background (CMB) radiation. Nevertheless, the specific mechanism that triggers the inflationary epoch is one of the most outstanding issues in contemporary particle physics and cosmology. As a result, the building of theoretical models that explain this early accelerating expansion of the Universe has exploded in recent years. The first main class of mechanisms that can lead to successful inflation is based on the introduction of a scalar field, while the second main class is obtained through gravitational modifications (for reviews see [26, 28, 46–48]). Conse- quently, inflation-related observations have provided significant insight to both modified gravity [49–53], as well as to particle physics model building. The literature on the latter is very extensive, particularly within the framework of supersymmetry [54–57], supergravity [58–61], theories of extra dimensions such as superstring and brane theories [62–64], and technicolor too [65]. Detailed lists of references on different theoretical constructions can be found in [26,28,46]. CHAPTER 3. COSMIC INFLATION 50

In trying to understand the above issues (often in the framework of a single theory) several problems have been encountered, including fine-tuning issues (tiny dimensionless constants) and large predictions for tensor fluctuations. In this respect, theories of scalar fields with non-canonical kinetic terms, as expected in supergravity and superstring theories, including the k-inflation subclass [40,66,67], were found to have significant advantages. These theories arise commonly in the framework of supergravity and string compactifications, which typically contain a large number of light scalar fields X (mod-

uli), whose dynamics are governed by a non-trivial moduli space metric Gij. As long as the moduli space metric is not flat, we generically expect non-canonical kinetic terms. Such effects could, but need not, be suppressed by the high scale of the corresponding Ultra-Violet physics (e.g. moduli masses, string scale), but they can still have significant cosmological consequences through the dynamics of the dilaton and moduli fields. Among their many advantages, non-canonical scalars satisfy in a more natural way the slow-roll conditions of inflation, since the additional effective friction terms in the equations of motion of the inflaton slow down the scalar field for potentials which would otherwise be too steep. Hence, the resulting tensor-to-scalar ratio is significantly reduced [68–85]. Moreover, models with non-canonical kinetic terms often allow for the kinetic term to play the role of dark matter and the potential terms to generate dark energy and inflation [86–89]. Additionally, note that in the inflation realization in the context of Galileon and Horndeski theories, the role of the non-canonical kinetic term is also crucial [90–95]. The form of the non-canonical terms can vary significantly, since there are many plausible models, including different ways to achieve compactification. The recent cosmological data, however, together with the requirement to avoid fine-tuning and unnatural solutions, severely constrain the available possibilities. On the other hand, an alternative way to improve the inflationary observables is by introducing an extra parameter as an exponent in the known potential forms, and thus affecting their steepness. In this way the dynamics of the scalar field can be additionally deformed, offering an alternative way to bring the tensor-to-scalar ratio to lower values without ruining the necessary spectral index [96–103]. One possible disadvantage of the above inflationary models, namely those with non- canonical terms and those with extra steepness parameter in the potential, is that the CHAPTER 3. COSMIC INFLATION 51

parameter values needed for acceptable observables are unnatural and hard to be justify from the ﬁeld-theoretical point of view. In particular, the non-canonical exponents need to be large, or the mass and potential parameters take trans-Planckian values. Hence, as we wil show in chapter 8, introducing a scalar ﬁeld with non-canonical kinetic terms on top of a deformed-steepness potential with an extra parameter [104], enhances the range of solutions and leads to very satisfactory observables, for natural sets of model parameters that we will identify and classify.

3.4.1 Non-Canonical Scalar Dynamics

The general action of a scalar ﬁeld which couples minimally to gravity, in the absence of any other matter sources is given by

Z 4 R = d x√ g + (X, φ) = EH + φ (3.63) S − 2κ L S S where the Lagrangian density (φ, X) is an arbitrary function of the scalar ﬁeld φ and L its kinetic term X which is given by

1 X = ∂ φ∂µφ. (3.64) 2 µ Varying the action (3.63) with respect to the metric yields the Einstein ﬁeld equations

φ Gµν = Tµν, (3.65)

φ where Tµν is the energy-momentum tensor of the scalar ﬁeld which is given by

φ ∂ T = L (∂µφ∂νφ) gµν . (3.66) µν ∂X − L Varying the action (3.63) with respect to the scalar ﬁeld φ leads to the equation of motion

∂ 1 ∂ L ∂µ √ g L = 0. (3.67) ∂φ − √ g − ∂(∂µφ) − In a homogeneous, isotropic spatially ﬂat FLRW Universe the ﬁeld φ is a function only of time i.e. , φ = φ(t), hence X = φ˙2/2 and the equation of motion (3.67) reduces to

∂ ∂2 ∂ ∂2 ∂ L + (2X) L φ¨ + (3H) L + φ˙ L φ˙ L = 0. (3.68) ∂X ∂X2 ∂X ∂X∂φ − ∂φ CHAPTER 3. COSMIC INFLATION 52

The expressions of the energy density ρφ and pressure pφ associated with the scalar ﬁeld are given by

∂ ρφ = L (2X) (3.69) ∂X − L

pφ = (3.70) L

Employing the FLRW metric, the Friedmann equations can be written as

8πG H2 = ρ (3.71) 3 φ ˙ 2 4πG H + H = (ρφ + 3pφ) . (3.72) − 3 where ρφ satisﬁes the conservation equation

ρ˙φ + 3H(ρφ + pφ) = 0. (3.73)

Note that the equation of motion for φ in (3.68) also follows from the conservation equation (3.73). The non-canonical Lagrangian density (X, φ), can be generally represented as L

(X, φ) = f(φ)F (X) V (φ) (3.74) L − where V (φ) is a self-interacting potential for the scalar ﬁeld φ, f(φ) is an arbitrary function of φ with f(φ) 0 and F (X) is an arbitrary function of X. The function f is ≥ required to explain various cosmological observations [105]. The general non-canonical Lagrangian (3.74) includes all the popular single scalar ﬁeld models:

K-essence: when V (φ) = 0 (X, φ) = f(φ)F (X) • ⇒ L Quintessence: when f(φ) = 1 and F (X) = X (X, φ) = X V (φ) • ⇒ L − Phantom: when f(φ) = 1 and F (X) = X (X, φ) = X V (φ) • − ⇒ L − −

We can see that quintessence and phantom fields are equivalent to a particular K-essence model. The idea of K-essence was firstly introduced as a possible model for inflation CHAPTER 3. COSMIC INFLATION 53

[40, 66] and later was considered as a possible model for dark energy [106–115]. Hence, Lagrangian (3.74) constitutes an alternative model of dark energy, yielding late time accelerated solutions, and it is well motivated from high-energy physics [116,117]. Several functional forms of F (X) can be considered and have been proposed so far [40, 66, 67, 87, 106, 118, 118–129]. In order to obtain the concrete cosmological dynamics we shall consider the non-canonical scalar ﬁeld model whose Lagrangian has the following form [67,119,130]

X α−1 (X, φ) = X V (φ), (3.75) L M 4 − X α−1 where in this case f(φ) = 1, F (X) = X M 4 , M has dimensions of mass while α is dimensionless. We see that when α = 1 the Lagrangian (3.75) reduces to the usual canonical scalar ﬁeld Lagrangian i.e. quintessence.

Substituting (3.75) into Eqs.(3.69),(3.70) for ρφ and pφ respectively, we obtain

X α−1 ρφ = (2α 1)X + V (φ) − M 4 X α−1 pφ = X V (φ) . (3.76) M 4 −

The two Friedmann equations (3.71) and (3.72) now become [67,123]

" # 8πG X α−1 H2 = (2α 1) X + V (φ) (3.77) 3 − M 4 X α−1 H˙ = 8πGαX , (3.78) − M 4

while Eq.(3.68) for the scalar ﬁeld equation of motion becomes

3Hφ˙ V 0(φ) 2M 4 α−1 φ¨ + + = 0, (3.79) 2α 1 α(2α 1) φ˙2 − − which reduces to the standard Klein-Gordon equation when α = 1,

φ¨ + 3Hφ˙ + V 0(φ) = 0. (3.80) CHAPTER 3. COSMIC INFLATION 54

3.4.2 Inﬂationary Parameters - Observables

In the non-canonical context we can also define the modified inflationary parameters as in the canonical case. Using Eq.(3.78) and Eq.(3.15) the first Friedmann equation (3.77) becomes

2α 1 8πG H2 1 − = V (φ). (3.81) − 3α 3 As concerns the second slow roll parameter, one ﬁnds ! φ¨ ηH = α , (3.82) − Hφ˙

and substituting (3.82) into (3.79) we obtain

2α 1 V 0 2M 4 α−1 3Hφ˙ 1 − δ = . (3.83) − 3α − α φ˙2 In the slow-roll regime the conditions 1, δ 1 imply the following relations | | between the slow-roll parameters and the inﬂaton potential

1 " 4 α−1 0 2α# 2α−1 1 3M MplV H V = ' α V √2V α ηH ηV V = (2ΓV V ) , (3.84) ' − 2α 1 − − where the parameter

V (φ)V 00(φ) Γ = (3.85) V 0(φ)2 plays a signiﬁcant role in inﬂationary and quintessence model building. When α = 1 Eqs.(3.84), converge to the standard canonical expressions of 3.2 i.e. Eqs.(3.23) § The slow-roll assumption also leads to

1 M θV 0(φ) 2α−1 φ˙ = θ pl 2M 4α−1 , (3.86) − α√3 √V where θ = +1 when V 0(φ) > 0 and θ = 1 when V 0(φ) < 0. Eq.(3.86) reduce to the − standard one of the canonical case when α = 1, i.e. Eq.(3.22). The scalar power spectrum (3.38) in the non-canonical slow-roll regime reads as CHAPTER 3. COSMIC INFLATION 55

1 " ! # 2α−1 1 α6α 1 V (φ)5α−2 S(k) = 2 4(α−1) 14α−8 0 2α (3.87) P 72π cs µ Mpl V (φ) where the speed of sound is determined from (3.32) and for the Lagrangian (3.75) has the form

1 c2 = . (3.88) s 2α 1 − It is easy to see, that for α > 1, cs < 1.

3.4.3 Some Examples Of Non-Canonical Inﬂationary Models

Chaotic Inﬂation

Let us consider the Lagrangian (3.75),

X α−1 = X V φ. (3.89) L M 4 − Following the steps of 3.3.2, we obtain the following expression for the value of the scalar § ﬁeld when inﬂation ends

1  α−1  γ(2α−1) 4(α−1) 4−n ! 2α φe µ 3Mpl n =   (3.90) Mpl α V0 √2

where

2α + n(α 1) M γ = − , µ = . (3.91) 2α 1 Mpl − Now by using the relation of the number of e-folds (3.27), we obtain the scalar ﬁeld as a function of the e-folds N

1 1 φ(N) γ n γ = D1 Nγ + , (3.92) Mpl 2 where

1  α−1 2α−1 4(α−1) 4−n ! n µ 6Mpl D1 =   . (3.93) α V0 CHAPTER 3. COSMIC INFLATION 56

The scalar power spectrum in this case reads as

φ γ+n (k) = A (3.94) S S M P pl aH=csk where

1  3α−2 2α−1 α ! 1 α6 V0 AS = 2  2α 4(α−1) 4−n  . (3.95) 72π cS n µ Mpl With these at hand we can determine the scalar spectral index in the non-canonical case for the Lagrangian (3.75) as

γ + n nS = 1 2 , (3.96) − 2Nγ + n and its running as

1 2 αS = n (nS 1) . (3.97) −1 + γ − For α = 1 Eqs.(3.96),(3.97), reduces to the standard result for large ﬁeld inﬂationary models with a canonical kinetic term, namely Eqs.(3.51),(3.53). Turning now to the tensor power index (3.40) we get

2n nT = , (3.98) −2Nγ + n while for the tensor-to-scalar ratio (3.44) we ﬁnd

1 16n r = . (3.99) √2α 1 2Nγ + n − As in the case of the canonical scalars we consider N = 60 and substitute n = 2 in Eq.(3.46. For the scalar spectral indexes and tensor-to-scalar ratios, in these cases, we obtain

Quadratic potential V (φ) = 1 m2φ2: • 2

4 nS = 1 (3.100) − 2N + 1 and CHAPTER 3. COSMIC INFLATION 57

1 16 r = . (3.101) √2α 1 Nγ + 1 − As we see the above result for the scalar spectral index nS does not depend upon the value of the α-attractor, so for this potential its value for N = 60 is identical in the canonical and non-canonical case.

Quartic potential V (φ) = 1 λφ4: • 4

γ + 4 nS = 1 (3.102) − Nγ + 2 and

1 32 r = . (3.103) √2α 1 Nγ + 2 − Since γ varies from γ = 2 (α = 1) to γ = 3 (α 1) for N = 60 we ﬁnd

nS 0.962, α 1. (3.104) ' Comparing the result (3.61) in the canonical case with the result (3.104) in the non- canonical case, we see that the the non-canonical scalars with Lagrangian of the form (3.75) improve the inﬂationary observables in agreement with observations.

The Exponential Potential

In the non-canonical frame, for the exponential potential (3.62), the slow-roll parameter (3.84) now becomes

1 " # 2α−1 1 3M 4 α−1 = (3.105) V αqα V (φ)

1 where for α = 1 the above equation reduces to the canonical one i.e. V = q . In contrast with the canonical case, there is a successful exit for α > 1, as the inﬂaton rolls down its

potential and the slow-roll parameter evolves from V 1 to V 1. ' For the exponential potential the scalar power spectrum (3.38) has the following expression CHAPTER 3. COSMIC INFLATION 58

1  3α−2 2α−1 α ! r 1 α(3q) V0 3α 2 2 φ S(k) = 2  4(α−1) 4  exp − , P 72π cS µ M − 2α 1 q Mpl pl − (3.106) and consequently, the scalar spectral index and tensor to scalar ratio for the exponential potential are

3α 1 n = 1 2 − , α > 1, (3.107) S − 2α 1 + 2N(α 1) − −

16√2α 1 r = − , α > 1, (3.108) 2α 1 + 2(α 1) − − respectively. It can be shown that for α > 1 the exponential potential can be accommo- dated by observations [72]. As we will show in chapter 8, assuming a single scalar ﬁeld with non-canonical kinetic terms on top of a generalized exponential potential with an extra parameter [104], enhances the range of solutions and leads to even more satisfactory values of the observables, for natural sets of model parameters. Chapter 4

Dark Matter

Without a doubt, the Standard Model (SM) of particle physics [131–133], is the corner- stone of our knowledge of all known elementary particles and their gauge interactions (excluding gravity). Nearly 50 years after the remarkable works of Higgs [134, 135] and independently of Englert and Brout [136], the last piece of the puzzle was ﬁnally unveiled in 2012 by LCH at CERN [137, 138]. As we mentioned in Chapter 2, ordinary matter constitutes only 15% of the total amount of matter of the Universe. The remaining 85% consists of this unknown kind of matter i.e. dark matter, which also constitutes about 27% of the total energy of the Universe. Consequently, despite the success of the SM, it is by now clear, that a more fundamental theory must exist, whose low-energy realization should coincide with the SM.

4.1 Extensions of the Standard Model

Several theories have been proposed as extensions of the SM. Two of the more prevalent are:

Supersymmetry (SUSY): The SM of particle physics provides a fundamental • distinction between bosons and fermions. In the context of SM bosons are the mediators of interactions while fermions are the constituents of matter. SUSY [139] provides a complete symmetry between bosons and fermions, thus providing a sort of uniﬁed picture of interactions and matter. Furthermore, it provides a possible solution to the hierarchy problem and last but not least, it provides an excellent

59 CHAPTER 4. DARK MATTER 60

dark matter candidate in terms of its lightest stable particle (LSP), the neutralino.

Universal Extra Dimensions (UED): Models of UED appear as an extra spa- • tial dimensional extensions of the SM of particle physics, following an early work of Kaluza-Klein [140], who extended to four the number of spatial dimensions to include Electromagnetism into a geometric theory of gravitation. Models of UED, such as the ADD model (Arkan, Dimopoulos, Dvali), attempt to solve the hierarchy problem but furthermore due to propagation of the particles of the SM in the extra dimensions, provide a viable DM candidate, namely, the lightest Kaluza-Klein particle (LKP), which corresponds to the ﬁrst excitations of the particles of the SM.

4.2 Evidence For Dark Matter

Historically, dark matter has not been an unknown concept in the scientific community. In 1884, Kelvin concluded that many of our stars, perhaps the majority of them, may be dark bodies. This conclusion came from the estimation of the mass of the galaxy, which he determined to be different from the mass of visible stars. This conclusion was followed later by various other physicists such as Henri Poincarè,Jacobus Kapteyn and Jan Oort. In 1933, Fritz Zwicky, while studying the motions of galaxies in the Coma Cluster, noticed that the ratio of its mass to its brightness and number of galaxies was about 400 times more than was visually observable [141]. This unseen/missing matter he called “dark matter”.

4.2.1 Galaxy Rotation Curves

The most direct evidence for dark matter comes from the study of the circular velocities of stars and gas as a function of their distance from the galactic center i.e. the observations of the rotation curves of galaxies. This evidence arises from the fact that although the rotation curve rises from the centre outward, at a suﬃciently large radius it invariably becomes ﬂat, even though most of the light is enclosed within this radius. A typical graph of the observed galactic rotation curve of spiral Messier 33 is shown in Fig.4.1. In the context of Newtonian gravity, the circular velocity of the stars is given by CHAPTER 4. DARK MATTER 61

Figure 4.1: Rotation curve of spiral galaxy Messier 33 (yellow and blue points with er- ror bars), and a predicted one from distribution of the visible matter (gray line). The discrepancy between the two curves can be accounted for by adding a dark matter halo surrounding the galaxy. Source [142]

r GM(r) υ(r) = (4.1) r where G is the gravitational constant, and M(r) is the enclosed mass of a sphere with radius r, which is given by the usual expression M(r) = 4π R ρ(r)r2dr. According to Gauss’s Law, beyond the optical disc M should remain constant with M(r) r and ∝ the velocity should be falling as υ(r) r−1/2. Thus, due to the fact that the velocity is ∝ approximately constant beyond the galactic disk, there must be a halo with

ρ(r) Mr−3 r−2. (4.2) ∝ ∼

4.2.2 Gravitational Lensing

Since DM interacts gravitationally, it drives ordinary matter i.e. gas and dust, to gather up and form our stars and galaxies. Although DM is an invisible form of matter, its CHAPTER 4. DARK MATTER 62 effect can be detected by observing how the gravity of massive galaxy clusters, which contain dark matter, bends and distorts the light of more-distant galaxies located behind the sources e.g. clusters [143]. This phenomenon is the so-called gravitational lensing. Through this effect, these objects act as astrophysical lenses creating a giant magnifying glass Fig.4.2. The distribution of mass in the lens can be determined by the size and shape of the image and then be compared with the visible mass. Several types of gravitational lensing exist. The simplest type of gravitational lensing occurs when the light of a distant galaxy is redirected around the dense core of a galaxy, where there is single concentration of matter around this core. This effect often produces multiple images of the background galaxy. In the case of almost perfect symmetry lensing, a circle of light is produced, called an Einstein ring. On the other hand, massive clusters of galaxies produce more complex gravitational lensing. In this case, the distribution of matter is never circularly symmetric and can be significantly lumpy. Background galaxies of the cluster are lensed by the cluster and their images often appear as short, thin “lensed arcs” around the outskirts of the cluster. The distribution of lensed images reflects the distribution of all matter, both visible and dark. The Bullet Cluster [144,145], is perhaps the most spectacular piece of evidence in favour of dark matter Fig.4.3. Thus, gravitational lensing probes the distribution of matter in galaxies and clusters of galaxies, and enables observations of the distant universe.

4.2.3 CMB Radiation

The early rapid expansion of the Universe caused by cosmic inflation, is believed to have smoothed out any lumpiness the Universe may have initially had, but quantum mechanical fluctuations introduced new ones i.e. tiny fluctuations of density at all length scales. These tiny fluctuations have grown with time due to gravity, eventually providing the seeds for the galaxies and galaxy clusters we see today. A map of the apparent temperature of the CMB across the sky thus gives a map of the density of all containing matter in the early Universe. A very precise, and consistent with the theoretical models prediction for the DM density is given by the CMB power spectrum, which is the amount of fluctuations in the CMB temperature spectrum at different angular scales on the sky, Fig.4.4. The peaks in CHAPTER 4. DARK MATTER 63

Figure 4.2: Galaxy cluster Abell 370, located about 4 billion light-years away, contains an astounding assortment of several hundred galaxies tied together by the mutual pull of gravity. Entangled among the galaxies are mysterious-looking arcs of blue light. These are actually distorted images of remote galaxies behind the cluster. These far-flung galaxies are too faint for Hubble to see directly. Instead, the gravity from the cluster acts as a huge lens in space that magnifies and stretches images of background galaxies like a funhouse mirror. Nearly 100 distant galaxies have multiple images caused by the lensing effect. The most stunning example is ”the Dragon,” an extended feature that is probably several duplicated images of a single background spiral galaxy stretched along an arc. Astronomers chose Abell 370 as a target for Hubble because its gravitational lensing effects can be used for probing remote galaxies that inhabited the early universe. Source: NASA, ESA, and J. Lotz and the HFF Team (STScI)

Fig.4.4 contain interesting physical signatures. The first peak determines the curvature of the universe. The next peak-ratio of the odd peaks to the even peaks-determines the reduced baryon density, while the third peak can be used to get information about the DM density. The best fit to the power spectrum as observed by the Planck 2018 mission [42], is a flat ΛCDM model, with baryonic and DM density

2 Ωbh = 0.02237 0.00015 (4.3) ± 2 ΩDM h = 0.1200 0.0012 , (4.4) ± where

H h = 0 = 0.6736 0.0054 (4.5) 100 km Mpc−1 s−1 ± CHAPTER 4. DARK MATTER 64

Figure 4.3: On the left panel is a color image from the Magellan images of the merging cluster 1E0657-558. On the right panel is the same image from Deep Chandra of the cluster. The X-ray brightness of the gas component is coded in yellow, red and blue colours, while in green contours in both panels is encoded the distribution of the gravitating mass, obtained from weak lensing reconstruction. Source: [145]

is the reduced Hubble parameter, with H0 = 67.36 0.54 the Hubble constant today. ±

Figure 4.4: Planck 2018 Cosmic Microwave Background radiation temperature anisotropy angular power spectra. Source: [42] CHAPTER 4. DARK MATTER 65 4.3 Constraints On Dark Matter

To be consistent with observations, there are some constraints about the nature of DM as a particle:

Neutral. The DM should be electromagnetically neutral, otherwise it would scatter • light and thus not be dark.

Non-baryonic. DM should be non-baryonic, otherwise it would radiate and thus • contribute to the baryonic component i.e. ordinary matter, measured from the Cosmic Microwave Backround anisotropy observations.

Relic abundance. The DM particle relic abundance, indicates the asymptotic • value of abundance of a species of a particle which it will reach after its “freeze- out”, i.e. to yield the observed cold dark matter.

Stable. DM does not decay. This means that the corresponding DM parti- • cle/particles must be stable and furthermore, its decay lifetime has to be obviously larger than the age of the Universe.

Cold/Warm and Non-relativistic. Agreement with observations requires that • the corresponding DM particle should be non-relativistic, to explain the large-scale structure of the Universe. In this context hot dark matter is relativistic, while cold dark matter is non-relativistic and warm dark matter starts to behave as non- relativistic at low temperatures [146–148].

4.4 Dark Matter Candidates

In this section we will brieﬂy review some of the most studied DM candidates.

Massive Astrophysical Compact Halo Objects (MACHOs). MACHOs are • non-luminous, ultra-compact objects, that emit little or no radiation, i.e. dark or very faint objects, that are very diﬃcult to detect. It is believed that the Galaxy may in principle be ﬁlled with a large population of this kind. At least a fraction of this kind of objects are baryonic objects, e.g. black holes, neutron stars, faint CHAPTER 4. DARK MATTER 66

stars (red, white and brown dwarfs) unassociated planets etc, and can be detected through microlensing eﬀects, which occur when a massive object passes through the line of sight of a distant object, bending the light from the distant source around it, brightening and magnifying the image [149]. Although most MACHOs have been observed in the halo of the Milky Way [150–154], the expected number would be larger if MACHOs were to comprise all of the missing mass of galaxies. Therefore, MACHOs can constitute only around 20% of the DM of the Universe.

Primordial Black Holes (PBH). PBHs are hypothetical objects that formed • in the radiation-dominated era i.e. before the nucleosynthesis era, due to the gravitational collapse of important density fluctuations. PBHs lighter than 1014 g would have evaporated due to Hawking radiation [155], PBHs larger than 1014 g [150, 151, 156–162] remain unaffected by Hawking radiation. Such PBHs might have various cosmological and astrophysical consequences but perhaps their most exciting possible feature is that they could provide DM [163–165]. Since PBHs were formed before the nucleosynthesis era, they are classified as non-baryonic, and that gives PBHs dynamics similar to any other form of cold DM. After the great discovery of gravitational waves in 2016 by the gravitational-wave observatories LIGO and VIRGO [166] there has been a renewed interest in PBHs as DM [167–173].

Light Neutrinos. Light neutrinos are the only DM candidate provided by SM. The • P current lower limit on the neutrino masses is mν < 0.12 eV [2], which means that light neutrinos cannot account for all the DM. Furthermore, if neutrinos accounted for all the DM, the large scale structure in our Universe would significantly differ from the one observed, due to its dependence on the velocity of the DM particles at the epoch of the formation of the early time structures. More specifically, if the DM particles were non-relativistic during decoupling, i.e. cold DM, the large scale structure would display sharp features, while in the case that DM particles were relativistic [174], i.e. hot DM, a smooth large scale structure would expected. This apparent disagreement of the actual galaxy distribution with the predictions of the hot DM paradigm, excludes SM neutrinos as the dominant component of DM.

Sterile Neutrinos. As light neutrinos with masses of few keV, would be ruled • out as DM candidates, their right-handed counterparts i.e. sterile neutrinos [175], CHAPTER 4. DARK MATTER 67

would be acceptable. Sterile neutrinos are fermions that are a singlet under the SM gauge group and only interact with the SM via a Yukawa coupling with the left-handed lepton doublet and the Higgs doublet. In this context, the parameter space contains only two parameters, i.e. the sterile neutrino mass and the mixing angle between the active and the sterile neutrino. Sterile neutrinos could be warm or cold DM, due to existence of a lepton asymmetry [176–178], which resonantly enhances the dark matter production.

Axions. Axions are hypothetical elementary particles, introduced to explain the • lack of CP violation in the strong interaction [179, 180]. Axions interact strongly with matter and they are associated with a new U(1) symmetry (Peccei-Quinn (PQ) symmetry) [179], in which the axion ﬁeld, after its breaking in the very early Universe, can give birth to axions, through the QCD phase transition mechanism, at which free quarks where bound into hadrons, a bose condensate of axions form and these very cold particles would naturally behave as cold DM [181–183]. Although experiments have constrained the available mass parameter range [184], axions still remain viable DM candidates.

Weakly Interacting Massive Particles (WIMPs). WIMPs are hypothetical • particles and are probably the most well-motivated and most studied DM candidates. A WIMP is usually considered to be a particle with mass in the range from around 2 GeV up to 100 TeV [185–187]. WIMPs interact weakly with the known particles. WIMP-like particles are predicted by various extensions of SM e.g. R- parity conserving supersymmetry (LSP), KK-parity conserving UED (LKP) and T-parity in little Higgs theories which provides a dark matter candidate in terms of its lightest T-odd particle (LTP). This renders them phenomenologically attractive since they could lie within the reach of current searches, including the LHC.

Various other DM candidates are Q-balls [188, 189], mirror particles [190–194], Charged Massive Particles (CHAMPs) [195], D-matter [196], etc. Chapter 5

Dark Energy

It’s been already two decades since observational data of two independent projects, the Supernovae Type Ia (SNIa) [8] and the High-redshift Supernova Search Team [9], showed that the present Universe is expanding at an accelerating rate. As dark matter has been used to describe the weak interaction of a non-relativistic (pressureless) matter with the ordinary (standard) matter, the source for this late-time acceleration of the Universe was dubbed dark energy (DE). In this context, this exotic form of energy plays the role of a force which exerts a negative, repulsive pressure and by counteracting the gravitational force leads to the accelerated expansion of the Universe. The observational data [8] have shown that dark energy constitutes about 68% of the present energy of the Universe, which still remains a great unsolved puzzle.

5.1 Evidence For Dark energy

5.1.1 Luminosity Distance

An important quantity, which helps us to describe the evolution of the Universe is the redshift parameter. This is related to the fact that light emitted by an object becomes red- shifted due to the expansion of the Universe. The relation between the redshift parameter and the scale factor is

a 1 + z = 0 . (5.1) a

68 CHAPTER 5. DARK ENERGY 69

Figure 5.1: This diagram reveals changes in the rate of expansion since the Universe’s birth 13.8 billion years ago. The more shallow the curve, the faster the rate of expansion. The curve changes noticeably about 7.5 billion years ago, when objects in the Universe began ﬂying apart as a faster rate. Astronomers theorize that the faster expansion rate is due to a mysterious, dark force that is pulling galaxies apart. Source: NASA, Ann Feild (STScI)

Additionally, for any theoretical model one can calculate the predicted dimensionless

luminosity distance dL(z), using the predicted evolution of the Hubble rate H(z) as

−1 Z z 0 d dL(z) dz H(z) = = dL(z) (1 + z) 0 (5.2) dz 1 + z ⇒ ≡ 0 H(z )

If the luminosity distance dL(z) can be measured observationally, then it can be used to determine the expansion rate of the universe. Furthermore, rewriting Eq.(2.45) in a more compact form as

2 2 X (0) 3(1+wi) H = H0 Ωi (1 + z) , (5.3) i where subscript (i) denotes each component and superscript (0) denotes its present value, the luminosity distance in a ﬂat geometry is given by

(1 + z) Z z dz0 dL = . (5.4) H q (0) 0 0 P 0 3(1+wi) i Ωi (1 + z ) Fig.5.2 shows the luminosity distance as determined by Eq.(5.4), and it is easy to see that it becomes larger when the cosmological constant is present. CHAPTER 5. DARK ENERGY 70

5.0

W (0) (d) (a) L = 0 (c) W (0) (b) L = 0.3 (b) 4.0 W (0) (c) L = 0.7 W (0) (d) L = 1 (a) 3.0 L d 0 H 2.0

1.0

0.0 0 0.5 1 1.5 2 2.5 3 z

Figure 5.2: The luminosity distance for a two component ﬂat universe (non-relativistic

ﬂuid with wm = 0 and cosmological constant with wΛ = 1). Source: [197] − 5.1.2 High-Redshift Supernovae Ia

The observation of luminosity distances of high redshift supernovae provides the most

direct evidence of the current acceleration of the Universe. The luminosity distance dL is related to the apparent magnitude m of a source with an absolute magnitude M via the relation [198,199]

d (z) m(z) M = 5 log L + 25. (5.5) − 10 Mpc When a white dwarf star exceeds the mass of the Chandrasekhar limit1 it explodes, and a type Ia SuperNova (SN Ia) is formed, which can be observed. SN Ia have a common absolute magnitude M independent of the redshift z i.e. their formation mechanism is the same and does not depend on their location in the Universe. Thus, the apparent magnitude m and the redshift z can be determined observationally and depend only upon the observable objects.

1The Chandrasekhar limit is the maximum mass of a stable white dwarf star. The currently accepted value of the Chandrasekhar limit is about 1.4 M

CHAPTER 5. DARK ENERGY 71

5.1.3 The Age Of The Universe

The existence of dark energy is crucially important to solve the cosmic age problem. This emerges from the fact that the age of the Universe, in the absence of a cosmological constant, is smaller than the the age of the oldest stellar populations. Let us denote the

age of the Universe by t0 and the age of the oldest stellar populations by ts. The age of the Universe needs to satisfy the lower bound: t0 > 11 12 Gyr [200, 201]. Recent − +0.13 observational data produce a best ﬁt value for the age of the Universe of t0 = 13.73−0.17 Gyrs [202]. From Friedmann equation (2.31) we obtain

2 2 (0) (0) −2 H = H [Ω Ω (a/a0) 0 Λ − k (0) −3 (0) −4 +Ωm (a/a0) + Ωr (a/a0) ] , (5.6)

(0) 2 2 where Ωk = k/(a0H0 ). In terms of the redshift z, the age of the Universe is given by

Z t0 Z ∞ dz t0 = dt = 0 0 H(1 + z) Z ∞ dz = . (5.7) (0) (0) 2 (0) 3 (0) 4 1/2 0 H0(1 + z)[Ω Ω (1 + z) + Ωm (1 + z) + Ωr (1 + z) ] Λ − k (0) 2 In the absence of the cosmological constant and setting Ωr = 0, the age of the Universe is given by

1 Z ∞ dz t0 = q . (5.8) H0 0 2 (0) (1 + z) 1 + Ωm z

(0) (0) (0) For a ﬂat Universe i.e. Ω = Ωm 1 = 0 Ωm = 1, we have k − ⇒

2 t0 = . (5.9) 3H0

Hence, Eq.(5.9) gives t0 = 8 10 Gyr, which does not satisfy the stellar age bound − t0 > 11 12 Gyr. − 2The radiation term can approximately be neglected due to the fact that the radiation dominated epoch is much shorter than the total age of the Universe. CHAPTER 5. DARK ENERGY 72

In the presence of the cosmological constant Eq.(5.7) now gives

1 Z ∞ dz t0 = q H0 0 (0) 3 (0) (1 + z) Ωm (1 + z) + ΩΛ

 q (0)  1 2 1 + ΩΛ = q ln  q  . (5.10) H0 (0) (0) 3 ΩΛ Ωm

In this case t0 = 13.1 Gyr. Hence, this satisﬁes the constraint t0 > 11 12 Gyr coming − from the oldest stellar populations. Thus, the presence of the cosmological constant Λ solves the age-crisis problem.

5.1.4 CMB and Large-Scale Structure (LSS)

As already stated the CMB supports the ideas of a dark energy dominated Universe. Independently from CMB, LSS [203,204] also gives crucial details about the size, density, and distribution of the observed structure. According to recent observational data [2, 42], Ω(0) = 0.0006 0.0019, which is con- k ± sistent with a ﬂat Universe, and Ω(0) = 0.6847 0.0073. Fig. 5.3 shows the conﬁdence Λ ± regions from SN Ia, CMB and large-scale galaxy clustering.

Recent observational data shows that the Universe is dark energy dominated with ΩΛ ' 0.7 and Ωm 0.3. '

5.2 Dark Energy Candidates

In this section we will brieﬂy review some of the most studied DE candidates.

5.2.1 Cosmological Constant

The simplest candidate for dark energy is the cosmological constant Λ. As previously mentioned the cosmological constant was ﬁrstly introduced by Einstein himself to achieve a static Universe. From the particle physics point of view, the cosmological constant naturally arises as an energy density of the vacuum. But despite its simplicity, it is diﬃcult to explain why the energy scale of Λ is so much larger (10121 times) than the CHAPTER 5. DARK ENERGY 73

3 No Big Bang

Supernovae

1 SNAP Target Statistical Uncertainty CMB Boomerang expands forever 0 y (cosmological constant) Maxima lapses eventuall vacuum energy density recol closed

Clusters flat -1 open

0 1 2 3 mass density

Figure 5.3: 68.3%, 95.4%, and 99.7% confidence regions of the Ωm and ΩΛ plane, constrained from the observations of SN Ia, CMB and large-scale galaxy clustering. Source: [205] observed dark energy density, if it originates from the vacuum energy density. This is the so-called cosmological constant problem [206]. We have already presented the Einstein field equations and the corresponding cosmological dynamical equations at 2.2.3 and 2.2.4, from which it is clear that Λ contributes § § negatively to the pressure term and hence exhibits a repulsive effect, which is a crucial ingredient for the observed acceleration of the Universe. In order to reproduce the cosmic acceleration today, we require that Λ should be of the order of the square of the present Hubble parameter value H0 i.e.

Λ H2 = (2.1332h 10−42 GeV)2 . (5.11) ≈ 0 ×

This corresponds to a critical density ρΛ,

2 Λmpl −47 4 ρΛ = 10 GeV . (5.12) 8π ≈ The vacuum energy density can be estimated as CHAPTER 5. DARK ENERGY 74

74 4 ρvac 10 GeV , (5.13) ≈ which is about 121 orders of magnitude larger than the observed value given by Eq. (5.12). This cosmological constant fine-tuning problem was present even before the observational discovery of the accelerated expansion of the Universe. Several attempts have been made to find a solution to the above problem such as adjustment mechanisms [207, 208], anthropic considerations [209–211], quantum gravity [212–214], higher-dimensional gravity [215, 216], supergravity [217, 218], string theory [219–223], vacuum fluctuations of the energy density [224], etc.

5.2.2 Dark Energy Scalar-Field Models

If the cosmological constant problem were to be solved in a way that it completely vanished, we would need to find alternative models of DE. There are two approaches for the construction of DE models. The first approach is based on modified matter models in which the energy-momentum tensor on the r.h.s. of the Einstein equations contains an exotic matter source with a negative pressure. The second approach is based on modified gravity models in which the Einstein tensor on the l.h.s. of the Einstein equations is modified. In what follows we will review some of the most studied DE models.

5.2.3 Quintessence

Quintessence is described by a single scalar field φ minimally coupled to gravity and has already been studied in 3.1. § From Eq. (3.11) we find that the Universe accelerates for φ˙2 < V (φ). This means that one requires a flat potential to give rise to an accelerated expansion. The equation of state for the field φ is given by Eq.(3.12) i.e.

p φ˙2 2V (φ) wφ = = − . (5.14) ρ φ˙2 + 2V (φ) CHAPTER 5. DARK ENERGY 75

5.2.4 K-essence

K-essence models rely on Quintessence models but in this case the accelerated expansion arises out of modifications to the kinetic energy of the scalar fields. We have already studied K-essence models in 3.4.1. § The equation of state of the K-essence field is given by

1 X wφ = − . (5.15) 1 3X − For wφ < 1/3, we have the condition X < 2/3, which gives rise to an accelerated − expansion of the Universe.

5.2.5 Tachyon ﬁeld

Rolling tachyon condensates, as a class of string theories. Tachyon fields has an equation of state with a value ranging between 1 and 0 [225]. The tachyon fields can also act as − a source of DE depending upon the form of the tachyon potential [108,226–229]. The action of the tachyon field on a non-BPS D3-brane is given by

Z 4 p S = d x V (φ) det(gab + ∂aφ∂bφ) . (5.16) − − The equations of motion for the tachyon ﬁeld are

2 8πGV (φ) H = q , (5.17) 3 1 φ˙2 − 8πG V (φ)φ˙ H˙ = , (5.18) 2 q − 1 φ˙2 − and

φ¨ 1 dV + 3Hφ˙ + = 0 (5.19) 1 φ˙2 V dφ − From Eq. (5.18) an accelerated expansion occurs for φ˙2 < 2/3. The equation of state of the tachyon ﬁeld is written as CHAPTER 5. DARK ENERGY 76

p ˙2 wφ = = φ 1 . (5.20) ρ −

5.2.6 Phantom ﬁeld

Quintessence, K-essence and tachyon fields correspond to an equation of state w 1. ≥ − The case where the equation of state is less than 1 is typically attributed due to some − form of phantom dark energy. The simplest explanation for the phantom DE is a scalar field with a negative kinetic energy [230]. The action of the phantom field minimally coupled to gravity is given by

Z 1 S = d4x√ g ( φ)2 V (φ) , (5.21) − 2 ∇ −

where the sign of the kinetic term is opposite compared to the action of Quintessence models for an ordinary scalar ﬁeld. The equation of state of the phantom ﬁeld is written as

p φ˙2 + 2V (φ) wφ = = . (5.22) ρ φ˙2 2V (φ) − Then an accelerated expansion occurs for φ˙2/2 < V (φ).

5.2.7 Chaplygin gas

Chaplygin gas [231] is a dark energy model that can lead to the acceleration of the Universe at late times whose pressure has the simple form

A p = , (5.23) − ρ where A is a positive constant. For tachyon ﬁelds p = V 2(φ)/ρ. Hence, the Chaply- − gin gas can be regarded as a special case of a tachyon ﬁeld with a constant potential. Consequently, the equation of state for this class of models is given by

A w = . (5.24) −ρ2 CHAPTER 5. DARK ENERGY 77 5.3 The Fate Of The Universe - Future Time Singu- larities

Some of the dark energy models predict the existence of exotic cosmological singularities, involving divergences of the scalar spacetime curvature and/or its derivatives. These singularities can be either geodesically complete [232–235] (geodesics continue beyond the singularity and the Universe may remain in existence) or geodesically incomplete [236, 237] (geodesics do not continue beyond the singularity and the Universe ends at the classical level). They appear in various physical theories such as superstrings [238], scalar field quintessence with negative potentials [239], modified gravities and others [235, 240, 241]. Violation of the cosmological principle (isotropy-homogeneity) by some cosmological models (e.g. modified gravity [242], quantum effects [243]), has been shown to eliminate or weaken both geodesically complete and incomplete singularities [244–263]. Geodesically incomplete singularities include the Big-Bang [264], the Big-Rip [265,266] where the scale factor diverges at a finite time due to infinite repulsive forces of phantom dark energy, the Little-Rip [267] and the Pseudo-Rip [268] singularities where the scale factor diverges at a infinite time and the Big-Crunch [234, 235, 239, 269–271] where the scale factor vanishes due to the strong attractive gravity of future evolved dark energy, as e.g. in quintessence models with negative potential. Geodesically complete singularities include SFS (Sudden Future Singularity) [240], Big-Brake singularity [272] (a subclass of the SFS singularities, characterized by a full stop of the expansion with finite scale factor, vanishing energy density and diverging deceleration and pressure), FSF (Finite Scale Factor) [273,274], BS (Big-Separation) and the w-singularity [275, 276]. In these singularities, the cosmic scale factor remains finite but a scale factor’s derivative diverges at a finite time. The singular nature of these “singularities” amounts to the divergence of scalar quantities involving the Riemann

a¨ a˙ 2 k tensor and the Ricci scalar R = 6 a + a2 + a2 , for the FRW metric, where a(t) is the cosmic scale factor [277]. Despite the divergence of the Ricci scalar, the geodesics are well defined at the time of the singularity. The Tipler and Krolak [278, 279] integrals of the Riemann tensor components along the geodesics are indicators of the strength of these singularities and remain finite in most cases. The Tipler integral [278] is defined as CHAPTER 5. DARK ENERGY 78

Z τ Z τ 0 0 00 i 00 dτ dτ R0j0(τ ) (5.25) 0 0 | | while the Krolak integral [279] is deﬁned as

Z τ 0 i 0 dτ R0j0(τ ) (5.26) 0 | | i where τ is the affine parameter along the geodesic and R0j0 is the Riemann tensor. The components of the Riemann tensor are expressed in a frame that is parallel transported along the geodesics. If the scale factor’s first derivative is finite at the singularity, both integrals are finite (even if the second derivative of the scale factor diverges), since the Riemann tensor components involve up to second order derivatives of the scale factor. If, however, the first derivative of the finite scale factor diverges, then it is easy to see from the above integrals that only the Tipler integral is finite at the singularity, while the Krolak integral diverges. This implies an infinite impulse on the geodesics, which dissociate all bound systems at the time of the singularity [233,280]. The singularities that lead to the divergence of the above integrals are defined as strong singularities [281,282]. The divergence of the scale factor and/or its derivatives leads to divergence of scalar quantities like the Ricci scalar and thus to different types of singularities or ‘cosmological milestones’ [277]. However geodesics do not necessarily end at these singularities and if the scale factor remains finite they are extended beyond these events [241] even though a diverging impulse may lead to dissociation of all bound systems in the Universe at the

time ts of these events [280]. Thus singularities can be classiﬁed [283] according to the behaviour of the scale factor

a(t) and/or its derivatives at the time ts of the event or equivalently according to the energy density and pressure of the content of the universe at the time ts. A classiﬁcation of such singularities and their properties is shown in Table 5.1. A particularly interesting type of singularity is the Sudden Future Singularity [240], which involves violation of the dominant energy condition ρ p , and divergence of the ≥ | | cosmic pressure of the Ricci Scalar and of the second time derivative of the cosmic scale factor. The scale factor can be parametrized as

t m t q a(t) = (as 1) + 1 1 , (5.27) ts − − − ts CHAPTER 5. DARK ENERGY 79

where m, q, ts are constants to be determined, as is the scale factor at the time ts and 1 < q < 2. The idea of such a finite time singularity and the above asymptotic form of a(t) was first introduced in [284], where it was used to show that closed FRW universes satisfying the strong energy condition ρ+3p 0, do not always recollapse. An interesting ≥ feature of such singularities includes the possibility of a quasi-isotropic solution which is part of the general solution of the Einstein equations, and approaches a late-time sudden singularity where the density, expansion rate, and metric remain finite. This solution has no equation of state and is characterised by nine independent arbitrary spatial functions [285]. In addition to (5.27) there are other similar parametrizations of the scale factor for sudden future singularities, which are also applicable for big-bang, big-rip, sudden future, finite scale factor and w-singularities [286,287].

For the range 1 < q < 2, Eq. (5.27) indicates that a, a˙ and ρ remain finite at ts. However, from pressure and the continuity equations it follows that p, ρ˙ anda ¨ become infinite. Thus, when the first derivative of the scale factor is finite at the singularity, but the second derivative diverges (SFS singularity [240]), the energy density is finite but the pressure diverges. SFS singularities (p + a¨ ) violate only the dominant → ∞ → −∞ energy condition (DEC): ρ p while respecting all other energy conditions (null energy ≥ | | condition (NEC): ρ + p 0, weak energy condition (WEC): ρ 0, ρ + p 0, strong ≥ ≥ ≥ energy condition (SEC): ρ + 3p 0. ≥ Geodesically complete singularities where the scale factor behaves like Eq. (5.27), are obtained in various physical models such as anti-Chaplygin gas [288,289], loop quantum gravity [263], tachyonic models [253–255, 272], brane models [250, 290, 291] etc. Such singularities however have not been studied in detail in the context of the simplest dark energy models of quintessence and scalar-tensor quintessence (see however [292, 293] for a qualitative analysis in the case of inflation). A singularity of the GSFS type (see Table 5.1), involving a divergence of the third derivative of the scale factor, occurs generically in quintessence models with potential of the form

V (φ) = A φ n, A > 0, (5.28) | | with 0 < n < 1 and A a constant parameter. In Ref. [292], it was shown through a CHAPTER 5. DARK ENERGY 80 qualitative analysis, that the power law scalar potential leads to singularities at any scale factor derivative order larger than three, depending on the value of the power n. In particular, for k < n < k + 1, with k > 0, the (k + 2)th derivative of the scale factor diverges at the singularity. Quintessence models with the potential (5.28) constitute the simplest extension of ΛCDM with geodesically complete cosmic singularities that occur at the time ts when the scalar ﬁeld becomes zero (φ = 0).

Table 5.1: Classiﬁcation and properties of cosmological singularities

Name tsing a(ts) ρ(ts) p(ts)p ˙(ts) w(ts) T K Geodesically Big-Bang (BB) 0 0 finite strong strong incomplete ∞ ∞ ∞ Big-Rip (BR) ts finite strong strong incomplete ∞ ∞ ∞ ∞ Big-Crunch (BC) ts 0 finite strong strong incomplete ∞ ∞ ∞ Little-Rip (LR) finite strong strong incomplete ∞ ∞ ∞ ∞ ∞ Pseudo-Rip (PR) finite finite finite finite weak weak incomplete ∞ ∞ Sudden Future (SFS) ts as ρs finite weak weak complete ∞ ∞ Big-Brake (BBS) ts as 0 finite weak weak complete ∞ ∞ Finite Sudden Future (FSF) ts as finite weak strong complete ∞ ∞ ∞ Generalized Sudden Future (GSFS) ts as ρs ps finite weak strong complete ∞ Big-Separation (BS) ts as 0 0 weak weak complete ∞ ∞ w-singularity (w) ts as 0 0 0 weak weak complete ∞

In chapter 6 we extend the analysis of [292,293], using a proper generalized expansion ansatz for the scale factor and the scalar field close to the singularity. As we show, these extra terms of the generalized ansatz, dominate close to the singularity and cannot be ignored when estimating the Hubble parameter and the scalar field energy density. Thus, they are important when deriving the observational signatures of such singularities. Furthermore we extend the above analysis to the case of scalar tensor quintessence with the same scalar field potential.

5.4 Thermodynamical Approach To Dark Energy

The usual approach of constructing modiﬁed gravitational theories is to start from the Einstein-Hilbert action and add correction terms. The simplest extension is to replace the Ricci scalar R by a function f(R) [43, 48, 294, 295]. Similarly, one can proceed in CHAPTER 5. DARK ENERGY 81

constructing many other classes of modiﬁcation, such as f(G) gravity [296,297], Lovelock gravity [298, 299], Weyl gravity [300, 301] and Galileon theory [302–304]. Alternatively, one can start from the torsional formulation of gravity and build various extensions, such

as f(T ) gravity [305–307], f(T,TG) gravity [308,309], etc. On the other hand, there is a well-known conjecture that one can express the Einstein equations as the first law of thermodynamics [310–312]. In the particular case of cosmology one can express the Friedmann equations as the first law of thermodynamics applied in the whole universe, considered as a thermodynamical system bounded by the apparent horizon and filled with the matter and dark-energy fluids [313–316]. Reversely, one can apply the first law of thermodynamics in the whole universe, and extract the Friedmann equations. Although this procedure is a conjecture and not a proven theorem, it seems to work perfectly in a variety of modified gravities, as long as one uses the modified entropy relation that corresponds to each specific theory [316–325]. Nevertheless, note that in order to know the modified entropy relation of a modified gravity, ones needs to know this modified gravity a priori and investigate it in spherically symmetric backgrounds. In this sense the above procedure cannot provide new gravitational modifications, offering only a way to study their features.

5.4.1 Friedmann equations as the ﬁrst law of thermodynamics

We consider the Universe as a thermodynamical system, which is ﬁlled with the matter

perfect ﬂuid, with energy density ρm and pressure pm. Although it is not obvious what its “radius” should be, namely the length that forms its boundary, there is a consensus that one should use the apparent horizon [313,314,326]

1 r˜a = q , (5.29) 2 k H + a2

a˙ with H = a the Hubble parameter and dots denoting derivatives with respect to t. The apparent horizon is a marginally trapped surface with vanishing expansion, deﬁned

ij in general by the expression h ∂ir∂˜ jr˜ = 0 (which implies that the vector r˜ is null ∇ or degenerate on the apparent horizon surface) [327]. For a dynamical spacetime, the apparent horizon is a causal horizon associated with the gravitational entropy and the surface gravity [327–329]. Finally, note that in ﬂat spatial geometry the apparent horizon CHAPTER 5. DARK ENERGY 82

becomes the Hubble one. The crucial point in the application of thermodynamics in cosmology, is that one can attribute to the universe horizon a temperature and an entropy that arise from the corresponding relations of black hole temperature and entropy respectively, but with the universe horizon, namely the apparent horizon, in place of the black hole horizon [310–312]. Concerning the black hole temperature, it is well known that for spherically symmetric geometry its relation does not depend on the underlying gravitational theory,

and it is just inversely proportional to the black hole horizon, namely T = 1/(2πrh) [330]. Hence, one can attribute to the universe horizon the temperature [312]

1 Th = , (5.30) 2πr˜a

independently of the gravitational theory that governs the universe. Concerning the black hole entropy, it is also known that its relation does depend on the underlying gravitational theory [312]. In the case of general relativity one obtains the usual Bekenstein-Hawking

2 relation S = A/(4G) (in units where ~ = kB = c = 1), where A = 4πrh is the area of the black hole and G the gravitational constant. Thus, in the case of a universe governed by general relativity, the horizon entropy will be just

1 S = A. (5.31) h 4G

Finally, a last reasonable assumption is that after equilibrium is established the Universe fluid acquires the same temperature with the horizon one, otherwise the energy flow would deform this geometry [331].3 As the universe evolves an amount of energy from the universe fluid crosses the horizon. During an infinitesimal time interval dt, the heat flow that crosses the horizon can be straightforwardly found to be [314]

δQ = dE = A(ρm + pm)Hr˜adt, (5.32) − 3Note that although this will certainly be the situation at late times, when the universe ﬂuid and the horizon will have interacted for a long time, it is not assured that it will be the case at early or intermediate times. However, in order to avoid applying non-equilibrium thermodynamics, which leads to mathematical complexity, the assumption of equilibrium is widely used [312–315, 323, 331]. Thus, we will follow this assumption and we will have in mind that our results hold only at late times of the universe evolution. CHAPTER 5. DARK ENERGY 83

2 with A = 4πra the apparent horizon area. On the other hand, the first law of thermodynamics states that dE = T dS. Since the temperature and entropy of the horizon − are given by (5.30) and (5.31) respectively, we find that dS = 2πr˜˙adt/G, with r˜˙a easily obtained from (5.29). Inserting the above into the first law of thermodynamics we finally acquire ˙ k 4πG(ρm + pm) = H . (5.33) − − a2 Additionally, assuming that the matter fluid satisfies the conservation equation

ρ˙m + 3H(ρm + pm) = 0, (5.34) inserting it into (5.33) and integrating we obtain

8πG 2 k Λ ρm = H + , (5.35) 3 a2 − 3

with Λ the integration constant, that plays the role of a cosmological constant. Interestingly enough, we saw that applying the first law of thermodynamics to the whole universe resulted to the extraction of the two Friedmann equations, namely Eqs. (5.33) and (5.35). The above procedure can be extended to modified gravity theories too, where as we discussed the only change will be that the entropy relation will not be the general relativity one, namely (5.31), but the one corresponding to the specific modified gravity at hand [316–325]. Nevertheless, we have to mention here that although the above procedure offers a significant tool to study the features and properties of various modified gravities, it does not lead to new gravitational modifications, since one needs to know the entropy relation, which in turn can be known only if a specific modified gravity is given a priori.

5.5 Modiﬁed Gravity

As we have already mentioned one can modify the l.h.s. of the Einstein equations and obtain modified gravity models, in which the origin of DE is identified as a modification of gravity. In modified gravity models one modifies the laws of gravity so that the late- time accelerated expansion of the Universe is realized without recourse to an explicit dark energy matter component. In what follows we will briefly review some basic characteristics CHAPTER 5. DARK ENERGY 84

Figure 5.4: Models of modiﬁed gravity. Source: Tessa Baker

of some of the most studied modiﬁed gravity models, such as f(R) gravity [43, 48, 294, 295], scalar-tensor theories [332] and Gauss-Bonnet gravity [296,297]. Similarly, one can proceed in constructing many other classes of modiﬁcation, such as Lovelock gravity [298, 299], Weyl gravity [300, 301] and Galileon theory [302–304]. Alternatively, one can start from the torsional formulation of gravity and build various extensions, such as f(T )

gravity [305–307], f(T,TG) gravity [308,309], etc Fig.5.4.

5.5.1 f(R) Gravity

f(R) gravity is one of the simplest modiﬁed gravities, in which the action reads as

Z 1 4 = d x√ gf(R) + m(gµν, Ψm) (5.36) S 2κ − S where f(R) is an arbitrary general function of the Ricci scalar, Ψm is the matter fields ∂f and F (R) = ∂R . We must note here that G is the modified gravitational constant, whose observed value is in general different from the usual one. CHAPTER 5. DARK ENERGY 85

We consider ﬂat FLRW spacetime where the Ricci scalar is given by

R = 6(2H2 + H˙ ) (5.37) and for matter sources we take into account non-relativistic matter and radiation, which satisfy the usual conservation equationsρ ˙m + 3Hρm = 0 andρ ˙r + 4Hρr = 0 respectively. The modiﬁed Friedmann equations in the case of the f(R) gravity model are

2 FR f ˙ 3FH = 8πG(ρm + ρr) + − 3HF (5.38) 2 − ˙ 4 ¨ ˙ 2F H = 8πG ρm + ρr + F HF. (5.39) − 3 − When F (R) = 1 and f(R) = R the modiﬁed Friedmann equations reduce to the standard ones.

5.5.2 Scalar-Tensor Theories

Scalar-tensor theories are probably the simplest example of modiﬁed gravity models and as such one of the most studied alternatives to GR. After the discovery of the acceleration of the Universe, they have been invoked by several authors [333–337] to generalize the cosmological constant and to explain the ﬁne-tuning and the coincidence problem. The action of scalar-tensor theories is given by

Z 4 1 1 2 = d x√ g f(φ, R) ζ(φ)( φ) + m(gµν, Ψm) (5.40) S − 2 − 2 ∇ S where f is a general function of the scalar ﬁeld φ. For convenience we choose units such that κ2 = 1. The above action includes a variety of scalar-tensor theories. For example, f(R) gravity corresponds to the choice f(φ, R) = f(R) and ζ = 0. Brans-Dicke theory corresponds to

the choice f = φR and ζ = ωBD/φ, where ωBD is the so-called Brans-Dicke parameter and other theories. Then the action (5.40) in the Jordan frame reads as

Z 4 1 1 2 2 = d x√ g F (φ)R (1 6Q )F (φ)( φ) U(φ) + m(gµν, Ψm) (5.41) S − 2 − 2 − ∇ − S CHAPTER 5. DARK ENERGY 86 where

" #−1/2 F F 3 F 2 ζ Q = ,φ = ,φ ,φ + (5.42) −2F −2F 2 − F F and

U V = . (5.43) F 2 The modiﬁed Friedmann equations in this case are

2 1 2 ˙2 ˙ 3FH = (1 6Q )F φ + U 3HF + ρm + ρr (5.44) 2 − −

˙ 2 ˙2 ¨ ˙ 4 2F H = (1 6Q )F φ F + HF ρm ρr (5.45) − − − − − 3

" ˙ # 2 ¨ ˙ F ˙ (1 6Q )F φ + 3Hφ + φ + U,φ + QF R = 0. (5.46) − 2F

5.5.3 Gauss-Bonnet Gravity

Gauss-Bonnet gravity is a modiﬁcation of GR that combines the Ricci and the Riemann tensors in a way that keeps the equations at second-order in the metric and does not necessarily give rise to instabilities. This is succeeded by a term, the Gauss-Bonnet term, which is a topological invariant quantity, which contributes to the dynamics in four dimensions provided that it is coupled to a dynamically evolving scalar ﬁeld. In units of κ = 1 the Gauss-Bonnet action is given by

Z 4 1 1 2 2 = d x√ g R ( φ) V (φ) f(φ)R + m(gµν, Ψm) (5.47) S − 2 − 2 ∇ − − GB S where

2 2 µν µναβ R = R 4RµνR + RµναβR (5.48) GB − is the Gauus-Bonnet term. The corresponding Friedmann equations in this case are

φ˙2 3H2 = + V (φ) + 24f φH˙ 3 + ρ + ρ , (5.49) 2 ,φ m r CHAPTER 5. DARK ENERGY 87

¨ ˙ 2 2 ˙ φ + 3Hφ Vφ + 24f,φH (H + H) = 0. (5.50) − Research

Chapter 6

Sudden Future Singularities in Quintessence and Scalar-Tensor Quintessence Models

In this work [338, 339] we derive analytically and numerically the cosmological solutions close to a future-time singularity for both quintessence and scalar-tensor quintessence models.

6.1 Sudden Future Singularities in Quintessence Mod- els

6.1.1 Evolution without perfect ﬂuid

Setting 8πG = 1, the most general action involving gravity, nonminimally coupled with a scalar ﬁeld φ and a perfect ﬂuid is

Z 1 1 µν 4 = F (φ)R + g φ;µφ;ν V (φ) + (fluid) √ gd x. (6.1) S 2 2 − L − In the special case where F (φ) = 1 and in the absence of a perfect ﬂuid, the action (6.1) reduces to the simple case of quintessece models without a perfect ﬂuid

Z 1 1 µν 4 = R + g φ;µφ;ν V (φ) √ gd x. (6.2) S 2 2 − −

90 CHAPTER 6. SFS IN SCALAR FIELD MODELS 91

The energy density and pressure of the scalar ﬁeld φ, may be written as

1 ˙2 1 ˙2 ρφ = φ + V (φ) and pφ = φ V (φ). (6.3) 2 2 − Variation of the action (6.2) assuming a power law potential (5.28) leads to the dynamical equations

1 3H2 = φ˙2 + V (φ) (6.4) 2

φ¨ = 3Hφ˙ An φ n−1Θ(φ) (6.5) − − | |

2H˙ = φ˙2, (6.6) − a˙ where H = a is the Hubble parameter, 0 < n < 1 and  1, φ > 0 Θ(φ) = (6.7)  1, φ < 0 − This class of quintessence models has been studied extensively focusing mostly on the cosmological effects and the dark energy properties that emerge due to the expected oscillations of the scalar field around the minimum of its potential [340–344]. In the present analysis we focus instead on the properties of the cosmological singularity that is induced as the scalar field vanishes periodically during its oscillations. For simplicity, we consider only the first time ts when the scalar field vanishes during its dynamical oscillations. Notice that the discontinuity of eq (6.5) is mild for 0 < n < 1 and is integrated out in the solution leading to no issues with instabilities or divergences. The dynamical evolution of the scalar field due to the potential shown in Fig. 6.1 may be qualitatively described as follows [293]: ˙ From eqs (6.4), (6.6), it follows that when t ts (φ 0) H, H remain finite and so → → does φ˙. But in eq. (6.5) there is a divergence of the term φn−1 for 0 < n < 1 and thus φ¨ as φ 0. H¨ also diverges at this point due to the divergence of φ¨, as follows → ∞ → by differentiating eq. (6.6). This implies that the third derivative of the scale factor diverges, and a GSFS occurs at this point (i.e. as, ρs, ps remain finite butp ˙ ). Thus, → ∞ CHAPTER 6. SFS IN SCALAR FIELD MODELS 92

V( )

1.5

1.0

0.5

-10 -5 5 10

Figure 6.1: Power law scalar ﬁeld potential V (φ) = A φ n Source: [338] | |

the constraints on the power exponents q, r of the diverging terms in the expansion of the

q r scale factor ( (ts t) ) and of the scalar ﬁeld ( (ts t) ) are 2 < q < 3 and 1 < r < 2 ∼ − ∼ − respectively (see eqs (6.10), (6.11) below). It has been shown in [345] that by choosing

q to lie in the intervals (N,N + 1) for N 2, where N Z+, a ﬁnite-time singularity ≥ ∈ occurs in which

dN+1a (6.8) dtN+1 → ∞ while

s d a + 0, for s N Z (6.9) dts → ≤ ∈ It may be shown that this allows for pressure singularities which are associated with divergence of higher time derivatives of the scale factor (divergence of the fourth-order derivative of the scale factor [345] when p ), in Friedmann solutions of higher-order → ∞ gravity (f(R)) theories [346]. In what follows we extend the above qualitative analysis to a quantitative level. In particular, we use a new ansatz for the scale factor and the scalar ﬁeld, containing linear

and quadratic terms of (ts t). These terms play an important role since they dominate − CHAPTER 6. SFS IN SCALAR FIELD MODELS 93 in the ﬁrst and second derivative of the scale factor as the singularity is approached. Thus, the new ansatz for the scale factor which generalizes (5.27), by introducing linear and quadratic terms in (ts t) is of the form −

2 q a(t) = as + b(ts t) + c(ts t) + d(ts t) , (6.10) − − − ... where b, c, d are real constants to be determined, and 2 < q < 3 so that a diverges at the GSFS. In [285], it has been shown that an asymptotic series for a general solution of the Einstein equations can be constructed near a sudden singularity. In our parametrization (6.10), we keep only the terms that can play an important role close to the singularity. The corresponding expansion of the scalar ﬁeld φ(t) close to the singularity is of the form

r φ(t) = f(ts t) + h(ts t) (6.11) − − where f, h, are real constants to be determined, and 1 < r < 2 so that φ¨ diverges at the singularity. Substituting eqs (6.10), (6.11) in eq. (6.5), we get the equation of the dominant terms

r−2 n−1 1(ts t) = 2(ts t) (6.12) A − A − where the 1, 2, denote constants, which may be expressed in terms of f, h and the A A constant A. Substituting the expressions (6.10), (6.11), (5.28) for a(t), φ(t) and V (φ) in the dynamical eqs. (6.4) and (6.6), it is straightforward to obtain relations among the expansion coeﬃcients as

b f = √6, (6.13) as

b2 c = , (6.14) −as

Af n−1 h = , (6.15) − n + 1

Ab√6f n−1 d = , (6.16) (n + 1)(n + 2) as has been veriﬁed by numerical solution of the dynamical equations. CHAPTER 6. SFS IN SCALAR FIELD MODELS 94

Also eq. (6.12) may be written explicitly as

r−2 n−1 n−1 hr(r 1)(ts t) = Anf (ts t) − − − −

Thus, the constants 1, 2 are A A

1 = hr(r 1) (6.17) A −

n−1 2 = Anf (6.18) A − Clearly, both the left and right-hand side of eq. (6.12) diverge at the singularity for 1 < r < 2 and 0 < n < 1. Equating the power laws of divergent terms we obtain

r = n + 1 (6.19)

Similarly, diﬀerentiation of eq. (6.6) with respect to t gives 2H¨ = 2φ˙φ¨, from which we − obtain an equation for the dominant terms using eqs (6.10), (6.11)

0 q−3 0 r−2 1(ts t) = 1(ts t) (6.20) A − A − 0 0 where the 1, 2 are constants, which may be expressed in terms of d, f, h. Eq. (6.20) A A may be written explicity as

dq(q 1)(q 2) q−3 r−2 − − (ts t) = fhr(r 1)(ts t) as − − − − 0 0 Thus, the constants 1, 2 are of the form A A

0 dq(q 1)(q 2) 1 = − − (6.21) A as

0 2 = fhr(r 1) (6.22) A − − The left and the right-hand side of eq. (6.20) diverge, and therefore, equating the power laws of diverging terms we obtain

q = r + 1. (6.23) CHAPTER 6. SFS IN SCALAR FIELD MODELS 95

Thus, using (6.19) and (6.23) we ﬁnd the exponent q in terms of n as

q = n + 2. (6.24)

8 n=0.5 ) t

( 7 ''' α 6 Log n=0.7 5

4 -11 -10 -9 -8 -7

Log(ts -t)

Figure 6.2: Numerical veriﬁcation of the q-exponent of scale factor a(t) for n = 0.5 and n = 0.7. The orange dashed line, denotes the analytical, while the blue line denotes the numerical solution. As expected the slopes for each n are identical. Source: [338]

Eqs (6.19), (6.24), are consistent with the qualitatively expected range of r, q, for 0 < n < 1.

The additional linear and quadratic terms in (ts t), in the expression of the scale − factor (6.10), play an important role in the estimation of the Hubble parameter and its derivative as the singularity is aproached. The relations between these coefficients can lead to relations between the Hubble parameter and its derivative close to the singularity, which in turn correspond to observational predictions that may be used to identify the presence of these singularities in angular diameter of luminosity distance data. For example, the coeficients b and c are related as eq. (6.14). The scale factor and its first and second derivative are

2 q a(t) = as + b(ts t) + c(ts t) + d(ts t) , (6.25) − − − CHAPTER 6. SFS IN SCALAR FIELD MODELS 96

7 n=0.5

) 5 t ( '' ϕ 4 Log 3 n=0.7

-16 -15 -14 -13 -12 -11 -10 -9

Log(ts -t)

Figure 6.3: Numerical veriﬁcation of the r-exponent of the scalar ﬁeld. As in Fig. 6.2 the slopes for each n are identical. Source: [338]

50 Scale factor evolution

40 n=0.7 n=0.8

) 30 n=0.9 t (

10 ts=7.46 0 ts=6.73 ts=8.4 0 2 4 6 8 10 t

Figure 6.4: Numerical (dashed) and analytical (line) time evolution of the scale factor, for n = 0.7, 0.8, 0.9. For each n the two solutions are consistent close to each singularity. Source: [338] CHAPTER 6. SFS IN SCALAR FIELD MODELS 97

2.0 Scalar field evolution 1.5

1.0 n=0.7 ) t

( 0.5 n=0.9 n=0.8 0.0

-0.5 ts=7.46

-1.0 ts=6.73 ts=8.4 0 2 4 6 8 10 t

Figure 6.5: Numerical (dashed orange line) and analytical (continuous blue line) time evolution of the scalar ﬁeld, for n = 0.7, 0.8, 0.9. For each n the two solutions are consistent close to each singularity. Source: [338]

q−1 a˙ = b 2c(ts t) dq(ts t) , (6.26) − − − − −

q−2 a¨ = 2c + dq(q 1)(ts t) . (6.27) − − Close to the singularity eqs (6.10), (6.26), (6.27) become

a(t) = as, (6.28)

a˙ = b, (6.29) − and

a¨ = 2c (6.30) respectively. Substituting eqs (6.28), (6.29), (6.30) into the Hubble parameter and its derivative we have CHAPTER 6. SFS IN SCALAR FIELD MODELS 98

-2 ) t (

''' -4 α Numerical Solutions for: n=0.7 (Red) n=0.8 (green) -6 n=0.9 (blue)

-8 1 2 3 4 5 6 7 8 t

Figure 6.6: Numerical solutions of the third time derivative of the scale factor for n = 0.7, 0.8, 0.9. Clearly the divergence of the third derivative of the scale factor occurs at the time of the singularity when the scalar ﬁeld vanishes (ts = 8.4 for n = 0.7, ts = 7.46 for

n = 0.8, ts = 6.73 for n = 0.9). Source: [338]

b H = (6.31) −as and

2 ˙ 2c b H = 2 (6.32) as − as Substituting eqs (6.31), (6.32) in eq. (6.14) it is easy to show that at the time of the

singularity ts we have

H˙ = 3H2. (6.33) − Equation (6.33) is identical to the corresponding equation describing a stiﬀ matter ﬂuid with ρ = p. In terms of the scale factor eq. (6.33) becomes

aHa + 3H(a) = 0

where Ha denotes derivative of H with respect to a. The solution is CHAPTER 6. SFS IN SCALAR FIELD MODELS 99

0.2

0.0

) -0.2 t ( '' ϕ -0.4 Numerical Solutions for: n=0.7 (Red) n=0.8 (green) -0.6 n=0.9 (blue)

-0.8 2 3 4 5 6 7 8 t

Figure 6.7: Numerical solutions of the second time derivative of the scalar ﬁeld for n = 0.7, 0.8, 0.9. Notice the divergence at the time of the singularity when the scalar ﬁeld vanishes (ts = 8.4 for n = 0.7, ts = 7.46 for n = 0.8, ts = 6.73 for n = 0.9). Source: [338]

H(a) = a−3.

The above equation expressed in terms of redshift z, is applicable at the singularity redshift zz and may be written as

3 H(zs) = (1 + zs) . (6.34)

This result constitutes an observationally testable prediction of this class of models, which can be used to search for such singularities in our past light cone.

6.1.2 Numerical analysis

It is straightforward to verify numerically the derived power law dependence of the scale factor and scalar ﬁeld as the singularity is approached. We thus solve the rescaled, with ¯ ¯ ¯ 2 the present day Hubble parameter H0 (setting H = HH0, t = t/H0, V = VH0 ), coupled system of the cosmological dynamical equations for the scale factor and for the scalar

ﬁeld (6.5) and (6.6). We assume initial conditions at early times (t t0) when the scalar CHAPTER 6. SFS IN SCALAR FIELD MODELS 100

˙ ﬁeld is assumed frozen at φ(ti) = φi and φ(ti) = 0 due to cosmic friction [347, 348]. At that time the initial conditions for the scale factor are well approximated by

"r # V (φ ) a(t ) = exp i t , (6.35) i 3 i

r " r # V (φ ) V (φ ) a˙(t )) = i exp i t . (6.36) i 3 3 i Taking the logarithm of the numerical solution corresponding to the third derivative of the scale factor (6.10) and to the second derivative of the scalar ﬁeld (6.11), we obtain

Fig.(6.2) and Fig.(6.3), which show these logarithms as functions of ts t close to the − singularity (continuous lines). On these lines we superpose the corresponding analytic ex- pansions (Eqs.(6.10), (6.11), dashed lines) which, close to the singularity, may be written as

... log[ a ] = log[ d q(q 1)(q 2)] + (q 3) log[(ts t)] (6.37) | | | | − − − − and

¨ log[ φ ] = log[ h r(r 1)] + (r 2) log[(ts t)]. (6.38) | | | | − − − In the plots of eqs (6.37), (6.38) (dashed lines) we have used the predicted values of the exponents (eqs (6.19) and (6.24)) and the analytically predicted values for the coeficients d and h. We underline the good agreement between the slopes of the analytically predicted curves and the corresponding numerical results, which confirms the validity of the power law ansatz (6.10), (6.11), and the values of the corresponding exponents (6.19), (6.24)). We have also verified this agreement by obtaining the best fit slopes of the numerical solutions of Fig.6.2, Fig.6.3 deriving the numerically predicted values of the exponents q and r. These numerical best fit values, along with the corresponding analytical predictions, are shown in Table 6.1 for n = 0.5 and n = 0.7, indicating good agreement between the analytical and numerical values of the exponents. In Fig.6.4, Fig.6.5 we show the time evolution (numerical and analytical) of the scale factor and the scalar field respectively. The two curves, for each n, are consistent close to each singularity. In Fig.6.6 and Fig.6.7 we demonstrate numerically the divergence of the CHAPTER 6. SFS IN SCALAR FIELD MODELS 101

third derivative of the scale factor and of the second derivative of the scalar ﬁeld. The divergence occurs at the time of the singularity when the scalar ﬁeld vanishes i.e. φ = 0.

Numerical Analytical n r q r = n + 1 q = n + 2

0.5 1.5 0.0003 2.51 0.0007 1.5 2.5 ± ± 0.7 1.7 0.002 2.71 0.004 1.7 2.7 ± ± Table 6.1: Numerical and analytical values of the power exponents r, q. Clearly, there is consistency between numerical results and analytical expectations.

6.1.3 Evolution with a perfect ﬂuid

In the presence of a perfect ﬂuid, the action of the theory is obtained from the generalized action (6.1) with F (φ) = 1 as

Z 1 1 µν 4 = R + g φ;µφ;ν V (φ) + (fluid) √ gd x. (6.39) S 2 2 − L − The corresponding dynamical equations are

3Ω 1 3H2 = m0 + φ˙2 + V (φ) (6.40) a3 2

φ¨ = 3Hφ˙ An φ n−1Θ(φ) (6.41) − − | |

3Ω 2H˙ = m0 φ˙2 (6.42) − a3 −

ρm0 3Ωm0 with ρm = a3 = a3 and Ωm0 = 0.3. The scale factor (6.10), in the presence of a perfect ﬂuid is now assumed to be of the form

m t 2 q a(t) = 1 + (as 1) + b(ts t) + c(ts t) + d(ts t) (6.43) − ts − − − 2 where m = 3(1+w) and w the state parameter, and the scalar field is of the same form (6.11), as in the case of the absence of the fluid term. As in the case of the previous section, from the dynamical equations (6.40), (6.42), H, H,˙ φ˙ still remain finite. Also in eq. (6.41) there is a divergence of the term φn−1 for 0 < n < 1 CHAPTER 6. SFS IN SCALAR FIELD MODELS 102

... and φ¨ as φ 0. The third derivative of the scale factor a also diverges due to the → ∞ → divergence of H¨ (diﬀerentiation of eq. (6.6)). Thus, the constraints for q, r are the same as in the absence of the ﬂuid, i.e. 2 < q < 3 and 1 < r < 2 respectively. Following the same steps, we rediscover the same values for the exponents i.e. eqs (6.19) and (6.24) which imply similar behaviour close to the singularity.

0.0

-0.1 n=0.5 n=0.7 n=0.9 ) t ( '' -0.2

-0.3

2 3 4 5 6 t

Figure 6.8: Plot of numerical solutions of the second time derivative of the scalar ﬁeld for n = 0.5, 0.7, 0.9. Source: [339]

Figure 6.8 shows the divergence of the second derivative of the scalar field at the time of the singularity. In Figs.(6.9), (6.10) we plot the numerically verified derived power law dependence (Eqs. (6.19), (6.24)) of the scalar field and the scale factor respectively, as the singularity is approached. It is clear that Eqs. (6.19), (6.24) are consistent with the qualitatively expected range of r, q, for 0 < n < 1. The relations among the expansion coefficients c, d, f, h in this case can be shown to be of the following form

" 2 #1/2 ((as 1)m b) ρm0 f = 6 − 2 − 2 3 , (6.44) as − as

2 ρm0 1 [(as 1)m b] c = 2 (as 1)m(m 1) − − , (6.45) 4as − 2 − − − as CHAPTER 6. SFS IN SCALAR FIELD MODELS 103

n=0.9 21

20 n=0.7 )

t 19 ( '''

α 18 n=0.5

Log 17

-10 -8 -6 -4 -2

Log(ts -t)

Figure 6.9: Numerical veriﬁcation of the q-exponent for three diﬀerent values of n (n = 0.5, n = 0.7 and n = 0.9) where the orange dashed line denotes the analytical, while the blue line denotes the numerical solution. As expected the slopes for each n are identical. Source: [339]

Af n−1 h = (6.46) − n + 1

n−1 r Af 2 ρm0 d = 6[(as 1)m b] 2 (6.47) (n + 1)(n + 2) − − − as and have been veriﬁed by numerical solution of the dynamical equations, as in the absence

of the fluid. For ρm0 = 0 all coefficients reduce to those of the no fluid case. As in the previous case, an interesting result arises from the derivation of the relation between the coefficients b, c (6.45). In the presence of the fluid term, the Hubble parameter

can be expressed in terms of the coeﬃcients as, b and m,

(as 1)m b H = − − (6.48) as and

2 ˙ (as 1)m(m 1) + 2c [(as 1)m b] H = − − − 2 − . (6.49) as − as CHAPTER 6. SFS IN SCALAR FIELD MODELS 104

7 n=0.5

5 ) t (

'' n=0.7

ϕ 4

Log 3

1 n=0.9

0 -16.0 -15.5 -15.0 -14.5 -14.0

Log(ts -t)

Figure 6.10: Plot of numerical verification of the r-exponent of the scalar field for three different values of n (n = 0.5, n = 0.7 and n = 0.9). As expected the slopes for each n are identical. Source: [339]

Substituting eqs (6.45), (6.48) in eq. (6.49) we ﬁnd

˙ 3Ωm0 2 H = 3 3H . (6.50) 2as − This may be written as

2 3Ωm0 aH(a)Ha + 3H (a) = 0 − 2a3 with solution

Ω 1 1 H2(a) = m0 [1 ] + . a3 − a3 a6

In terms of the singularity redshift zs this becomes

p 3 3 6 H(zs) = Ωm0(1 + zs) [1 (1 + zs) ] + (1 + zs) . (6.51) −

Clearly Eq.(6.50) reduces to Eq.(6.34) for ρm0 = 0. This result may be used as an observational signature of such singularities in this class of models. CHAPTER 6. SFS IN SCALAR FIELD MODELS 105 6.2 Sudden Future Singularities in Scalar-Tensor Quintessence Models

6.2.1 Evolution without a perfect ﬂuid

We now consider scalar-tensor quintessence models without the presence of a perfect

ﬂuid. The action of the theory is the generalized action (6.1), where (fluid) is ignored. L Therefore, it has the form

Z 1 1 µν 4 = F (φ)R + g φ;µφ;ν V (φ) √ gd x (6.52) S 2 2 − − We assume a nonminimal coupling linear in the scalar ﬁeld F (φ) = 1 λφ even though − our results about the type of the singularity in this class of models are unaﬀected by the particular choice of the nonminimal coupling. The dynamical equations are of the form

φ˙2 3FH2 = + V 3HF˙ (6.53) 2 −

¨ ˙ a¨ 2 (n−1) φ + 3Hφ 3Fφ + H + An φ Θ(φ) = 0 (6.54) − a | |

a¨ 2F H2 = φ˙2 + F¨ HF,˙ (6.55) − a − − d ˙ ˙ where Fφ = F . From Eq. (6.53), it is clear that H, φ, F, F all remain ﬁnite when φ 0 dφ → (t ts). However, in Eq. (6.54) there is a divergence of the term Vφ for 0 < n < 1 and → φ¨ as φ 0. This means that F¨ because of the generation of the second → ∞ → → ∞ derivative of φ that leads to a divergence ofa ¨ in Eq. (6.55). The eﬀective dark energy density and pressure take the form [235,349]

˙2 φ 2 ˙ ρDE = + V 3FH 3HF (6.56) 2 − −

˙2 φ ˙ 2 ¨ ˙ pDE = V (2H 3H )F + F + 2HF. (6.57) 2 − − −

Thus ρDE remains ﬁnite in Eq. (6.56), while pDE in Eq. (7.8). Clearly, an SFS → ±∞ singularity (Table 5.1, see also [350]) is expected to occur in scalar-tensor quintessence CHAPTER 6. SFS IN SCALAR FIELD MODELS 106

models, as opposed to the GSFS singularity in the corresponding quintessence models. This result will be veriﬁed quantitatively in what follows. Using the ansatz (6.10), (6.11) in the dynamical eq. (6.55) we ﬁnd that the dominant terms close to the singularity are

q−2 r−2 1(ts t) = 2(ts t) (6.58) B − B −

where the 1, 2 are constants, which depend on the coeﬃcient d, h and the λ constant. B B In this case the dynamical equations lead to the following relations among the expansion coeﬃcients

3λb √3pb2(2 + 3λ2) f = , (6.59) − as ± as

1 d = λa h, (6.60) 2 s

b2 5 c = + λbf, (6.61) −as 4

Af n−1 h = . (6.62) 3 2 −(n + 1) 1 + 2 λ We notice that all coefficients except d, are the same as in the case without the presence of the fluid for λ = 0. The reason that the coefficient d differs in scalar-quintessence is because in this case the divergence occurs in the second, instead of the third derivative of the scale factor. Eq. (6.58) is written explicitly, keeping only the dominant terms

dq(q 1) q−2 r−2 2 − (ts t) = λhr(r 1)(ts t) as − − −

Thus, the constants 1, 2 are B B

dq(q 1) 1 = 2 − (6.63) B as

2 = λhr(r 1). (6.64) B − CHAPTER 6. SFS IN SCALAR FIELD MODELS 107

It immediately follows from eq. (6.58) that

q = r (6.65)

Similarly, substituting the ansatz (6.10), (6.11) in eq. (6.54) we ﬁnd that the dominant terms close to the singularity obey the equation

0 r−2 0 n−1 1(ts t) = 2(ts t) (6.66) B − B − 0 0 where the 1, 2 are constants, which depend on the coeﬃcient f and the constants B B A, λ. Similarly, Eq. (6.66) is written explicitly, keeping only the dominant terms

3 2 r−2 n−1 n−1 λ + 1 r(r 1)(ts t) = Anf (ts t) 2 − − − − 0 0 Thus, the constants 1, 2 are B B 0 3 2 1 = λ + 1 r(r 1) (6.67) B 2 −

0 n−1 2 = Anf . (6.68) B − Equating the exponents of the divergent terms we ﬁnd

r = n + 1, (6.69)

which leads to q = n + 1. (6.70)

The results (6.69) and (6.70) are consistent with the above qualitative discussion for the expected strength of the singularity. Thus in the case of the scalar-tensor theory we have a stronger singularity at ts, compared to the singularity that occurs in quintessence models. This is a general result, valid not only for the coupling constant of the form F = 1 λφ but also for other forms of F (φ)(e.g. F φr), because the second derivative − ∼ of F with respect to time, in the dynamical equations, will always generate a second derivative of φ with divergence, leading to a divergence ofa ¨. Using Eqs (6.53), (6.58), (6.65), (6.66) and (6.69), we calculate relations among the coeﬃcients c, d, f, h. The form of these relations is shown in the Appendix, and has been CHAPTER 6. SFS IN SCALAR FIELD MODELS 108

n=0.5 6

) 5 t ( '' α 4 Log 3 n=0.8 2

-15 -14 -13 -12 -11

Log(ts -t)

Figure 6.11: Numerical verification of the q-exponent of the scale factor for n = 0.5 and n = 0.8. The slopes for each n are identical, while the small difference is due to the coefficients. Source: [338] verified by numerical solution of the dynamical equations. Notice that all coefficients, except d, are the same as in the case without the perfect fluid for λ = 0. 1

6.2.2 Numerical analysis

We now solve the rescaled coupled system of the cosmological dynamical equations for the scale factor and for the scalar ﬁeld (6.54) and (6.55), using the present day Hubble ¯ ¯ ¯ 2 parameter H0 (setting H = HH0, t = t/H0, V = VH0 ). We assume initial conditions at ˙ early times (t t0) when the scalar ﬁeld is assumed frozen at φ(ti) = φi and φ(ti) = 0 due to cosmic friction in the context of thawing [347,348] scalar-tensor quintessence [351–353]. At that time the initial conditions for the scale factor are

s  V (φi) a(ti) = exp  ti , (6.71) 3Fi

1The coeﬃcient d diﬀers in scalar-quintessence since the divergence occurs in the second, instead of the third derivative of the scale factor. CHAPTER 6. SFS IN SCALAR FIELD MODELS 109

5 n=0.5

4 ) t ( ''

ϕ 3 Log 2 n=0.8

-13 -12 -11 -10

Log(ts -t)

Figure 6.12: Numerical verification of the r-exponent for n = 0.5 and n = 0.8. As in Fig. 6.11 the slopes for each n are identical, while the small difference is due to the coefficients. Source: [338]

s  s V (φi) V (φi) a˙(ti)) = exp  ti , (6.72) 3Fi 3Fi where Fi = 1 λφi. − Taking the logarithm of the second derivative of the scale factor (6.10) and of the scalar ﬁeld (6.11), we obtain

log[ a¨ ] = log[ d q(q 1)] + (q 2) log[(ts t)] (6.73) | | | | − − − and

¨ log[ φ ] = log[ h r(r 1)] + (r 2) log[(ts t)] (6.74) | | | | − − − The numerical veriﬁcation of the validity of Eqs (6.69), (6.70) has been performed similarly to the case of minimally coupled quintessence. In Fig.(6.11) and Fig.(6.12) we show the analytical and numerical solutions, for the logarithm of the diverging terms of the scale factor and the scalar ﬁeld respectively, as t ts from below. The log-plots of → CHAPTER 6. SFS IN SCALAR FIELD MODELS 110

30 14 12 25 10 8 ts=5.2 20 6

) 4 t =4.88 t =5.56

t s s (

α 15 3.5 4.0 4.5 5.0 5.5

0 0 2 4 6 8 10 t

Figure 6.13: Numerical (dashed) and analytical (line) time evolution of the scale factor, for n = 0.7 (red), 0.8 (green), and 0.9 (blue). For each n the two solutions are consistent close to each singularity. Source: [338]

2.0 0.4 0.2 1.5 0.0 -0.2

-0.4 ts=5.2

-0.6 ts=4.88 ts=5.56

) 1.0 t ( 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 ϕ

0.5

0.0

0 2 4 6 8 t

Figure 6.14: Plot of time evolution of the scalar ﬁeld, for n = 0.7 (red), 0.8 (green), and 0.9 (blue). Source: [338] CHAPTER 6. SFS IN SCALAR FIELD MODELS 111

2.5 Numerical Solutions for: n=0.7 (Red) n=0.8 (green) 2.0 n=0.9 (blue) 1.5

) 1.0 t ( ''

α 0.5

0.0

-0.5

-1.0 1 2 3 4 5 t

Figure 6.15: Numerical solutions of a¨ for n = 0.7, 0.8, 0.9. The divergence occurs at

ts = 5.56 for n = 0.7, ts = 5.2 for n = 0.8, ts = 4.88 for n = 0.9. Source: [338]

0.0

-0.1

-0.2 ) t (

'' -0.3 ϕ -0.4

-0.5 Numerical Solutions for: n=0.7 (Red) n=0.8 (green) -0.6 n=0.9 (blue)

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 t

Figure 6.16: Numerical solutions of φ¨ for n = 0.7, 0.8, 0.9. The divergence occurs at ts = 5.56 for n = 0.7, ts = 5.2 for n = 0.8, ts = 4.88 for n = 0.9. Source: [338] the diverging terms ofa ¨ and φ¨ are straight lines, indicating a power law behaviour with best ﬁt slopes as shown in Table 6.2, in good agreement with the analytical expansion expectations (eqs (6.69), (6.70). In Figs.6.13, 6.14 we show the time evolution (numerical CHAPTER 6. SFS IN SCALAR FIELD MODELS 112

and analytical) of the scale factor and the scalar field respectively. The two curves, for each n, are consistent close to each singularity. In Figs.(6.15), (6.16) we demonstrate numerically the divergence of the second derivarive of the scale factor and of the scalar field. As expected, the divergence occurs at the time of the singularity when the scalar field vanishes.

Numerical Analytical n r q r = n + 1 q = n + 1

0.5 1.5 0.0003 1.49 0.0002 1.5 1.5 ± ± 0.8 1.8 0.03 1.8 0.006 1.8 1.8 ± ± Table 6.2: Numerical and analytical values of the power-laws r, q. Clearly, there is consistency between numerical results and analytical expectations.

Using Eqs (6.73), (6.74), it is straightforward to obtain numerically the values of the parameters h of the scalar ﬁeld, as well as d of the scale factor, and compare with their analytically obtained values shown in the Appendix.

The quadratic term of (ts t), in the expression of the scale factor (6.10), is now − subdominant as the second derivarive of the scale factor diverges. The only additional

term of (ts t) that can play an important role in the estimation of the Hubble parameter, − is the linear term. Clearly, for the ﬁrst derivative of (6.10), as t ts from below, the → linear term dominates over all other terms, while the quadratic term is subdominant in the second derivative in the divergence of the q-term. Thus, in the case of the scalar-

tensor quintessence models H remain ﬁnite and dominated by the term b(ts t), while − ˙ H as t ts. → ∞ →

6.2.3 Evolution with a perfect ﬂuid

In the presence of a perfect fluid, the action is now the generalized action (6.1). The scale factor and the scalar field are of the form (6.43) and (6.11) respectively. The dynamical equations in the presence of a relativistic fluid become

3Ω φ˙2 3FH2 = m0 + + V 3HF˙ (6.75) a3 2 − CHAPTER 6. SFS IN SCALAR FIELD MODELS 113

¨ ˙ a¨ 2 φ + 3Hφ 3Fφ + H + Vφ = 0 (6.76) − a

a¨ 3Ω 2F H2 = m0 + φ˙2 + F¨ HF˙ (6.77) − a − a3 − dF ˙ ˙ where Fφ = . From Eq. (6.75), it is clear that H, φ, F, F all remain ﬁnite when φ 0 dφ → (t ts). However, in Eq. (6.76) there is a divergence of the term Vφ for 0 < n < 1 and → φ¨ as φ 0. This means that F¨ because of the generation of the second → ∞ → → ∞ derivative of φ that leads to a divergence ofa ¨ in Eq. (6.77). Clearly, an SFS singularity (Table 5.1) is expected to occur in scalar-tensor quintessence models, as opposed to the GSFS singularity in the corresponding quintessence models. Thus, the constraints on the power exponents q, r in this case are 1 < q < 2 and 1 < r < 2 respectively. From the above dynamical equations, using the same parametrizations (6.43), (6.11) for the scale factor and the scalar ﬁeld respectively and keeping only the dominant terms, the values for r and q are

q = r (6.78)

r = n + 1, (6.79)

which leads to

q = n + 1. (6.80)

In figures (6.17), (6.18) we illustrate the numerically verified derived power law dependence Eqs (6.79), (6.80) of the scalar field and the scale factor respectively, as the singularity is approached. Figs. (6.19), (6.20), depict the divergence of the second derivative, of both the scale factor and the scalar field, at the time of the singularity. The results (6.79) and (6.80) are consistent with the above qualitative discussion for the expected strength of the singularity. Thus, in the case of the scalar-tensor theory, we have a stronger singularity at ts, as compared to the singularity that occurs in quintessence models. This is a general result, valid not only for the coupling constant of the form F = 1 λφ but also for other forms of F (φ)(e.g. F φr), because the second derivative − ∼ CHAPTER 6. SFS IN SCALAR FIELD MODELS 114

4 n=0.2 ) t ( n=0.4 '' 3 α

Log 2 n=0.6

-15 -14 -13 -12 -11

Log(ts -t)

Figure 6.17: Numerical veriﬁcation of q-exponent of the scale factor in the scalar-tensor case, for three diﬀerent values of n (n = 0.2, n = 0.4 and n = 0.6). Source: [339]

6 n=0.2 ) t ( n=0.4 '' 5 ϕ

Log 4 n=0.6 3

-13 -12 -11 -10

Log(ts -t)

Figure 6.18: Numerical verification of r-exponent of the scalar field in the scalar-tensor case, for three different values of n (n = 0.2, n = 0.4 and n = 0.6). Source: [339]

of F with respect to time, in the dynamical equations, will always generate a second derivative of φ with divergence, leading to a divergence ofa ¨. CHAPTER 6. SFS IN SCALAR FIELD MODELS 115

-0.2

-0.4

-0.6

0.8 n=0.6 n=0.4 n=0.2 ) - t ( '' -1.0

-1.2

-1.4

-1.6 1 2 3 4 5 t

Figure 6.19: Plot of the second time derivative of the scale factor for n = 0.2, 0.4, 0.6. Source: [339]

n=0.6 n=0.4 n=0.2 -1

) - t ( '' -3

-4

-5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 t

Figure 6.20: Plot of the second time derivative of the scalar ﬁeld for n = 0.2, 0.4, 0.6. The divergence of the scalar ﬁeld occurs at the time of the singularity. Source: [339]

The quadratic term of (ts t), in the expression of the scale factor (6.43), is now − subdominant as the second derivarive of the scale factor diverges. The only additional CHAPTER 6. SFS IN SCALAR FIELD MODELS 116

term of (ts t) that can play an important role in the estimation of the Hubble parameter, − is the linear term. Clearly, for the first derivative of (6.43), as t ts from below, the → linear term dominates over all other terms, while the quadratic term is subdominant in the second derivative, in the divergence of the q-term. Thus, in the case of the scalar- tensor quintessence models H remains finite and dominated by the term b(ts t), while − ˙ H as t ts. → ∞ → As in the quintessence case of the previous section, in the absence of the perfect fluid the strength of the singularity remains unaffected. This means that the evaluated relations of r and q, Eqs (6.79), (6.80) respectively, are exactly the same. Finally, as in the previous cases we use the relevant dynamical equation which in this case is Eq. (6.75), to obtain the relations among the expansion coefficients c, d, f, h as

s 2 2 b + m 3as(2 + 3λ )(b + m mas) 2ρm0 f = 3λ m 3 − − , (6.81) − as ± as

1 d = λa h, (6.82) 2 s

2 ρm0 1 [(as 1)m b] 5 c = 2 (as 1)m(m 1) − − λf[(as 1)m b] (6.83) 4as − 2 − − − as − 4 − − and

Af n−1 h = . (6.84) 3 2 −(n + 1) 1 + 2 λ

Notice that for ρm0 = 0, all coefficients reduce to the ones in the absence of the fluid. Comparing them with the coefficients of quintessence models, we see that for λ = 0 they reduce to them except for the coefficient d. This occurs because d is the coefficient of the scale factor’s diverging term. In quintessence models we have divergence of the third derivative of the scale factor, while in scalar-tensor models the second derivative of the scale factor diverges. Chapter 7

Modiﬁed Cosmology through non-extensive horizon Thermodynamics

In this work [354] we construct a modiﬁed cosmological scenario through the application of the ﬁrst law of thermodynamics, but using the generalized, nonextensive Tsallis entropy instead of the usual Bekenstein-Hawking one.

7.1 Tsallis entropy

In this subsection we brieﬂy review the concept of nonextensive, or Tsallis entropy [355–357]. As Gibbs pointed out already in 1902, in systems where the partition function diverges, the standard Boltzmann-Gibbs theory is not applicable, and large-scale gravitational systems are known to fall within this class. Tsallis generalized standard thermodynamics (which arises from the hypothesis of weak probabilistic correlations and their connection to ergodicity) to nonextensive ones, which can be applied in all cases, while still possessing standard Boltzmann-Gibbs theory as a limit. Hence, the usual Boltzmann- Gibbs additive entropy must be generalized to the nonextensive, i.e non-additive entropy (the entropy of the whole system is not necessarily the sum of the entropies of its sub- systems), which is named Tsallis entropy [355–359]. In cases of spherically symmetric systems as considered in this work, it can be written in compact form as [360]:

117 CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 118

α˜ S = Aδ, (7.1) T 4G 2 in units where ~ = kB = c = 1, where A L is the area of the system with characteristic ∝ length L, G is the gravitational constant,α ˜ is a positive constant with dimensions [L2(1−δ)] and δ denotes the non-additivity parameter.1 Under the hypothesis of equal probabilities the parameters δ andα ˜ are related to the dimensionality of the system [360] (in particular the important parameter δ = d/(d 1) for d > 1), however in the general case they remain − independent and free parameters. Obviously, in the case δ = 1 andα ˜ = 1, Tsallis entropy becomes the usual Bekenstein-Hawking additive entropy.

7.2 Modiﬁed Friedmann equations through the non-extensive ﬁrst law of thermodynamics

In subsection 5.4.1 we presented the procedure to extract the Friedmann equations from the first law of thermodynamics. This procedure can be applied in any modified gravity, as long as one knows the black hole entropy relation for this specific modified gravity. Hence, as we mentioned above, although it can be enlightening for the properties of various modified gravities, the thermodynamical approach does not lead to new gravitational modifications since one needs to consider a specific modified gravity a priori. In the present subsection however, we desire to follow the steps of subsection 5.4.1, but instead of the standard additive entropy relation to use the generalized, nonextensive, Tsallis entropy presented in subsection 7.1 above. Doing so we do obtain modified Friedmann equations, with modification terms that appear for the first time, and which provide the standard Friedmann equations in the case where Tsallis entropy becomes the standard Bekenstein-Hawking one. We start from the first law of thermodynamics dE = T dS, where dE is given by − − (5.32), T by (5.30), but we will consider that the entropy is given by Tsallis entropy (7.1).

2 In this case, and recalling that A = 4πr˜a we acquire

1 δ In [360] the nonextensive entropy relation is written as ST = γA , however we prefer to write is as in (7.1) for convenience. CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 119

δα˜ dS = (4π)δ r˜ 2δ−1r˜˙ dt. (7.2) 2G a a

Inserting everything in the ﬁrst law, and calculating r˜˙a from (5.29), we obtain

2−δ ˙ k (4π) G H a2 (ρm + pm) = δ − . (7.3) − α˜ 2 k δ−1 H + a2 Finally, inserting the conservation equation (5.34) and integrating, for δ = 2 we obtain 6

2−δ 2−δ ˜ 2(4π) G δ 2 k Λ ρm = H + , (7.4) 3˜α 2 δ a2 − 3˜α − where Λ˜ is an integration constant. Hence, the use of Tsallis entropy in the first law of thermodynamics, led to two modified Friedmann equations, namely (7.3) and (7.4), with modification terms that appear for the first time depending on three parameters out of which two are free. Let us elaborate on the obtained modified Friedmann equations. From now on we focus on the flat case, namely we consider k = 0, which allows us to extract analytical expressions. The investigation of the non-flat case is straightforward. We can re-write (7.3), (7.4) as

8πG H2 = (ρ + ρ ) (7.5) 3 m DE

˙ H = 4πG (ρm + pm + ρDE + pDE) , (7.6) − where we have deﬁned the eﬀective dark energy density and pressure as

( ˜ ) 3 δ−1 Λ 2 δ−1 δ 2(1−δ) ρDE = (4π) + H 1 α˜(4π) H , (7.7) 8πG 3 − 2 δ − n 1 δ−1 ˜ ˙ δ−1 2(1−δ) pDE = (4π) Λ + 2H 1 α˜(4π) δH −8πG − δ +3H2 1 α˜(4π)δ−1 H2(1−δ) . (7.8) − 2 δ − We can further simplify the above expressions by redeﬁning Λ (4π)δ−1Λ˜ and α ≡ ≡ (4π)δ−1α˜, obtaining CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 120

3 Λ 2 δ 2(1−δ) ρDE = + H 1 α H , (7.9) 8πG 3 − 2 δ −

1 ˙ 2(1−δ) 2 δ 2(1−δ) pDE = Λ + 2H 1 αδH + 3H 1 α H . (7.10) −8πG − − 2 δ − Thus, we can deﬁne the equation-of-state parameter for the eﬀective dark energy sector as

p 2H˙ 1 αδH2(1−δ) w DE = 1 − . (7.11) DE 2 αδ 2(1−δ) ≡ ρDE − − Λ + 3H 1 H − 2−δ In summary, in the constructed modified cosmological scenario, equations (5.34), (7.5) and (7.6) can determine Universe evolution, as long as the matter equation-of-state parameter is known. In particular, inserting (7.9), (7.10) into (7.6), we acquire a differential equation for H(t) that can be solved similarly to all modified-gravity and dark-energy models. Finally, as one can see, in the case δ = 1 and α = 1 the generalized Friedmann equations (7.5),(7.6) reduce to ΛCDM cosmology, namely

8πG Λ H2 = ρ + (7.12) 3 m 3

˙ H = 4πG(ρm + pm). (7.13) − We close this subsection by providing for completeness the equations for δ = 2. In this special case, integration of (7.3), instead of (7.4) results to

˜ G 2 k Λ ρm = ln H + . (7.14) 3˜α a2 − 6˜α Hence, in this case the two Friedmann equations (7.3) and (7.14), for k = 0, lead to the deﬁnitions

3 Λ 2 2 ρDE = + H 2α ln H (7.15) 8πG 3 − CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 121

1 2 2 ˙ 2α pDE = Λ+3H 6α ln H +2H 1 , (7.16) −8πG − − H2

and thus

˙ 2α pDE 2H 1 H2 wDE = 1 2 − 2 . (7.17) ≡ ρDE − − Λ + 3H 6α ln H −

7.3 Cosmological evolution

In this section we proceed to a detailed investigation of the modiﬁed cosmological scenarios constructed above. The cosmological equations are the two modiﬁed Friedmann equations (7.5) and (7.6), along with the conservation equation (5.34). In the general

case of a general matter equation-of-state parameter, wm pm/ρm, analytical solutions ≡ cannot be extracted, and thus one has to solve the above equations numerically. However, we are interested in providing analytical expressions too, and thus in the following we focus on the case of dust matter, namely wm = 0. As usual for convenience we introduce the matter and dark energy density parameters respectively as

8πG Ω = ρ (7.18) m 3H2 m 8πG Ω = ρ . (7.19) DE 3H2 DE

ρm0 In the case of dust matter, equation (5.34) gives ρm = a3 , with ρm0 the value of

the matter energy density at present scale factor a0 = 1 (in the following the subscript “0” marks the present value of a quantity). Therefore, in this case equation (7.18) gives

2 3 2 immediately Ωm = Ωm0H0 /a H . Combining this with the fact that Ωm + ΩDE = 1 we can easily extract that

√Ω H H = m0 0 . (7.20) p 3 a (1 ΩDE) − In the following we will use the redshift z as the independent variable, deﬁned as

1 + z = 1/a for a0 = 1. Thus, diﬀerentiating (7.20) we can obtain the useful expression CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 122

2 ˙ H 0 H = [3(1 ΩDE) + (1 + z)ΩDE], (7.21) −2(1 ΩDE) − − where a prime denotes derivative with respect to z. Inserting (7.9) into (7.19) and using (7.20) we obtain

1 δ−2 2 3 (2 δ) 2 3 Λ ΩDE(z) = 1 H Ωm0(1 + z) − H Ωm0(1+z) + . (7.22) − 0 αδ 0 3 This expression is the analytical solution for the dark energy density parameter

ΩDE(z), in a ﬂat universe and for dust matter. Applying it at present time, i.e at z = 0, we acquire

3αδ 2(2−δ) 2 Λ = H 3H Ωm0, (7.23) 2 δ 0 − 0 − which provides the relation that relates Λ, δ and α with the observationally determined

quantities Ωm0 and H0, leaving the scenario with two free parameters. As expected, for δ = 1 and α = 1 all the above relations give those of ΛCDM cosmology. Diﬀerentiating (7.22) we ﬁnd

3−δ (2 δ) Λ 1 δ−2 1 1 0 2 3 δ−2 ΩDE(z) = − 1 + 2 3 Ωm0H0 (1 + z) αδ 3 Ωm0H0 (1 + z) · αδ 2 2 Λ 3(δ 1)Ωm0H (1 + z) + (δ 2) . (7.24) · − 0 − 1 + z Hence, we can now calculate the other important observable, namely the dark-energy ˙ equation-of-state parameter wDE from (7.11), eliminating H through (7.21), and obtaining

h 2 3 i1−δ 0 H0 Ωm0(1+z) 3[1 ΩDE(z)] + (1 + z)ΩDE(z) 1 αδ { − } − 1−ΩDE (z) wDE(z) = 1 + (7.25) h 2 3 i1−δ − Λ[1−ΩDE (z)] αδ H0 Ωm0(1+z) [1 ΩDE(z)] H2Ω (1+z)3 + 3 1 2−δ 1−Ω (z) − 0 m0 − DE

0 where ΩDE and ΩDE are given by (7.22) and (7.24) respectively. Lastly, it proves convenient to introduce the decelaration parameter q 1 H˙ , which using (7.21) is found ≡ − − H2 to be CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 123

1 0 q(z) = 1 + 3[1 ΩDE(z)] + (1 + z)ΩDE(z) . (7.26) − 2[1 ΩDE(z)]{ − } − In summary, considering dust matter and ﬂat geometry we were able to extract analytical solutions for ΩDE(z) and wDE(z), for the modiﬁed, nonextensive cosmological scenarios of the present work. In the following two subsections we will investigate these in two distinct cases, namely when the explicit cosmological constant Λ is present and when it is absent.

7.3.1 Cosmological evolution with Λ = 0 6 We ﬁrst examine the case where the explicit cosmological constant Λ is present. In this case when δ = 1 and α = 1 we obtain ΛCDM cosmology, and thus we are interested in studying the role of the nonextensive parameter δ on the cosmological evolution.

We use relation (7.23) in order to specify the value of Λ that corresponds to Ωm0 0.3 ≈ in agreement with observations [361]. Moreover, in order to isolate the eﬀect of δ, we set α to its standard value, namely α = 1 (although for δ = 1 the parameter α is dimensionless, as we have mentioned for δ = 1 it acquires dimensions [L2(1−δ)] where for convenience we 6 use units where H0 = 1).

In Fig. 7.1 we depict ΩDE(z) and Ωm(z) = 1 ΩDE(z), as given by equation (7.22), − in the case where δ = 1.1. In Fig. 7.2 we present the corresponding evolution of wDE(z) according to (7.25). Finally, in Fig. 7.3 we present the deceleration parameter q(z) from (7.26). We mention that for transparency we have extended the evolution up to the far future, namely up to z 1, which corresponds to t . → − → ∞ As we observe, we acquire the usual thermal history of the universe, with the sequence of matter and dark energy epochs, with the transition from deceleration to acceleration taking place at z 0.45 in agreement with observations. Additionally, in the future the ≈ universe tends asymptotically to a complete dark-energy dominated, de-Sitter state. We mention the interesting behavior that although at intermediate times the dark-energy equation-of-state parameter may experience the phantom-divide crossing and lie in the phantom regime, at asymptotically large times it will always stabilize at the cosmological constant value 1. Namely, the de-Sitter solution is a stable late-time attractor, which − is a signiﬁcant advantage (this can be easily shown by taking the limit z 1 in → − CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 124

1.0 Ωm 0.8

0.6 Ω 0.4

0.2

ΩDE 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 z

Figure 7.1: The evolution of the nonextensive dark energy density parameter ΩDE (black- solid) and of the matter density parameter Ωm (red-dashed), as a function of the redshift z, for δ = 1.1 and α = 1 in units where H0 = 1. As we can see we acquire the usual thermal history of the Universe. Source: [354]

0.5

0.0 DE w -0.5

-1.0

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 z

Figure 7.2: The evolution of the corresponding dark-energy equation-of-state parameter wDE. Source: [354] CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 125

1.0

0.5

0.0 q

-0.5

-1.0

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 z

Figure 7.3: The evolution of the corresponding deceleration parameter q. Source: [354]

0 (7.22),(7.24) and (7.25), which gives ΩDE 1, Ω 0, and wDE 1, respectively). → DE → → − Let us now examine in detail the role of δ in the evolution, and in particular on wDE.

In Fig. 7.2 we depict wDE(z) for α = 1 and for various values of δ, including the value δ = 1 that reproduces ΛCDM cosmology. For each value of δ we choose Λ according to

(7.23) in order to obtain Ωm(z = 0) = Ωm0 0.3 at present, and obtain an evolution of ≈ ΩDE(z) and Ωm(z) similar to the graph of Fig. 7.1. In this way we can examine the pure effect of δ. Firstly, as we mentioned, for δ = 1 we obtain wDE = 1 = const., namely − ΛCDM cosmology. For increasing δ > 1, at earlier redshifts wDE acquires larger values, while on the contrary in the recent past, i.e at 0 z 0.8, wDE acquires algebraically ≤ . smaller values, which is also true for its present value wDE0. In all cases the universe experiences the phantom-divide crossing, and in the far future it approaches from below in a de-Sitter phase with wDE being 1. On the other hand, for decreasing δ < 1 the − behavior of wDE(z) is the opposite, namely it initially lies in the phantom regime, it then crosses the 1 divide from below, being quintessence-like at present, and finally it − asymptotically tends to 1 from above. − In summary, we can see that the nonextensive parameter δ, that lies at the core of the modified cosmology obtained in this work, plays an important role in giving dark energy a dynamical nature and bringing about a correction to ΛCDM cosmology. We mention that CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 126

1 δ=1.3

δ=1.2 0 δ=1.1

δ=1 DE -1 w

-2

δ=0.9 -3 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 z

Figure 7.4: The evolution of the dark-energy equation-of-state parameter wDE as a function of the redshift z, for α = 1 in units where H0 = 1, and various values of the nonextensive parameter δ. For each value of δ we choose Λ according to (7.23) in order to obtain Ωm(z = 0) = Ωm0 0.3 at present, and acquire an evolution of ΩDE and Ωm ≈ similar to the graph of Fig. 7.1. Source: [354] in all the above examples we kept the parameter α ﬁxed, in order to maintain the one- parameter character of the scenario. Clearly, letting α vary too, increases the capabilities of the model and the obtained cosmological behaviors.

7.3.2 Cosmological evolution with Λ = 0

In the previous subsection we investigated the scenario of modiﬁed Friedmann equations through nonextensive thermodynamics, in the case where the cosmological constant is explicitly present. Thus, we studied models that possess ΛCDM cosmology as a subcase, and in which the nonextensive parameter δ and its induced novel terms lead to corrections to ΛCDM paradigm. In the present subsection we are interested in studying a more radical application of the scenario at hand, namely to consider that an explicit cosmological constant is not present and let the model parameters δ and α to mimic its behavior and produce a CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 127

cosmology in agreement with observations. In the case Λ = 0, relations (7.9), (7.10) become

3 2 δ 2(1−δ) ρDE = H 1 α H (7.27) 8πG − 2 δ − 1 2 δ 2(1−δ) ˙ 2(1−δ) pDE = 3H 1 α H + 2H 1 αδH , (7.28) −8πG − 2 δ − − while (7.22) reads

1 δ−2 (2 δ) 2 3δ−1 ΩDE(z) = 1 − Ωm0H (1 + z) . (7.29) − αδ 0 However, the important simpliﬁcation comes from expression (7.23), that relates Λ

and α with the observationally determined quantities Ωm0 and H0. In particular, setting

Λ = 0 leads to the determination of parameter α in terms of Ωm0 and H0, namely

(2 δ) 2(δ−1) α = − Ωm0H , (7.30) δ 0 leaving δ as the only free model parameter. Note that sinceα ˜ > 0 in (7.1), i.e α > 0, from (7.30) we deduce that the present scenario is realized for δ < 2. Thus, inserting (7.30) into (7.29) leads to the simpliﬁed expression

3(δ−1) ΩDE(z) = 1 Ωm0(1 + z) δ−2 . (7.31) − Finally, inserting (7.30) and (7.31) into (7.25) and (7.26) gives respectively

(δ 1) h 3(δ−1) i−1 wDE(z) = − 1 Ωm0(1 + z) (δ−2) , (7.32) (2 δ) − − and

2δ 1 q(z) = − . (7.33) 2(2 δ) − We stress here that in this case exact ΛCDM cosmology cannot be obtained for any parameter values, and thus one should suitably choose δ in order to obtain agreement with observations. Note that in the standard extensive choice δ = 1 we obtain a trivial universe with ΩDE(z) = 1 Ωm0 = const. and wDE(z) = 0. − CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 128

1.0 Ωm 0.8

0.6 Ω 0.4

0.2

ΩDE 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 z

Figure 7.5: Plot of the evolution of ΩDE (black-solid) and of the matter density parameter

Ωm (red-dashed), as a function of the redshift z, for δ = 0.6 and Λ = 0 in units where

H0 = 1. Source: [354]

0.0 δ=1 δ=0.9 -0.5 δ=0.8 δ=0.7 -1.0 δ=0.6

DE -1.5 w

-2.0 δ=0.5 -2.5 δ=0.4 -3.0 -1.0 -0.5 0.0 0.5 1.0 1.5 z

Figure 7.6: Plot of the evolution of wDE as a function of the redshift z, for Λ = 0 in units where H0 = 1, and various values of the nonextensive parameter δ. Source: [354] CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 129

From the analytical expression (7.31) we can see that we acquire the thermal history of the universe, with the sequence of matter and dark energy epochs and the onset of late- time acceleration. Furthermore, in the future (z 1) the universe tends asymptotically → − to complete dark-energy domination. Additionally, as can be seen from expression (7.32),

the asymptotic value of wDE in the far future is not necessarily the cosmological constant

value 1. In particular, we deduce that for 1 δ < 2 wDE 0 as z 1, while − ≤ → → − for δ < 1 wDE (δ 1)/(2 δ) as z 1. Hence, the case δ < 1 is the one that → − − → − exhibits more interesting behavior in agreement with observations, and we observe that

for decreasing δ the wDE(z) tends to lower values. We close this subsection mentioning that according to the above analysis the cosmological behavior is very eﬃcient for low redshifts and up to the far future, despite the fact that an explicit cosmological constant is absent. However, as can be seen from (7.29),

for high redshifts the behavior of ΩDE(z) is not satisfactory, since as it stands this ex-

pression leads to either early-time dark energy or to the unphysical result that ΩDE(z) becomes negative. In order to eliminate this behavior and obtain a universe evolution in agreement with observations at all redshifts one needs to include the radiation sector too, which indeed can regulate the early-time behavior. This is performed in the next subsection.

7.3.3 Cosmological evolution including radiation

In this subsection for completeness we extend the scenario of modified cosmology through nonextensive horizon thermodynamics, in the case where the radiation fluid is also present. First of all, in the case where extra fluids are considered in the universe content, the thermodynamical procedure of 5.4.1 is applicable in exactly the same way, with the only § straightforward addition being that in Eq. (5.32) one should add the energy densities and pressures of all universe fluids [312–315, 323, 331]. Hence, if we allow for a radiation

ﬂuid, with energy density ρr and pressure pr, and repeat the analysis of subsection 7.2, the Friedmann equations (7.5), (7.6) become

8πG H2 = (ρ + ρ + ρ ) (7.34) 3 m r DE ˙ H = 4πG (ρm + pm + ρr + pr + ρDE + pDE) , (7.35) − CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 130

with ρDE, pDE still given by (7.9), (7.10), and wDE by (7.11). We proceed by introducing the radiation density parameter as

8πG Ωr ρr, (7.36) ≡ 3H2

and thus the ﬁrst Friedmann equation becomes Ωr + Ωm + ΩDE = 1. Similarly to the analysis of section 7.3, in order to extract analytical expressions we consider that the

matter ﬂuid is dust, namely wm = 0. In the case where radiation is present we still have 2 3 2 that Ωm = Ωm0H0 /a H , however (7.20) now extends to

√Ω H H = m0 0 , (7.37) p 3 a (1 ΩDE Ωr) − − while (7.21) reads

H2 3Ω + 4Ω (1 + z) (1 + z)Ω0 H˙ = m0 r0 + DE , (7.38) − 2 Ωm0 + Ωr0(1 + z) (1 ΩDE) − since for dust matter we have

4 Ωr0(1 + z) [1 ΩDE(z)] Ωr(z) = 3 − 4 . (7.39) Ωm0(1 + z) + Ωr0(1 + z)

Cosmological evolution with Λ = 0 6 Let us ﬁrst investigate the case where Λ = 0. Inserting (7.9) into (7.19) and using (7.37) 6 we ﬁnd that (7.22) extends to

2 3 4 ΩDE(z) = 1 H0 Ωm0(1 + z) + Ωr0(1 + z) − 1 δ−2 (2 δ) 2 3 4 Λ − H Ωm0(1+z) +Ωr0(1+z) + . (7.40) · αδ 0 3

This expression is the analytical solution for the dark energy density parameter ΩDE(z), in a ﬂat universe and for dust matter, in the case where radiation is present. Applying it at present time, i.e at z = 0, we acquire

3αδ 2(2−δ) 2 Λ = H 3H (Ωm0 + Ωr0) , (7.41) 2 δ 0 − 0 − CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 131

which provides the relation that relates Λ, δ and α with the observationally determined quantities Ωm0,Ωr0 and H0, leaving the scenario with two free parameters. As expected, for δ = 1 and α = 1 all the above relations give those of ΛCDM cosmology with radiation sector present. Diﬀerentiating (7.40) we ﬁnd

0 1 1 −1 Ω (z) = (z) δ−2 (z) (z) (z) 1 , (7.42) DE A B αδ B C −

where

2 2 3 (z) = H [3Ωm0(1 + z) + 4Ωr0(1 + z) ], A 0

2−δ 2 3 4 Λ (z) = H [Ωm0(1 + z) + Ωr0(1 + z) ] + B αδ 0 3

and

2 3 4 (z) = H [Ωm0(1 + z) + Ωr0(1 + z) ]. C 0 ˙ Hence, wDE(z) is calculated from (7.11), but now eliminating H through (7.38), we obtain

1−δ h A(z) 0 i h C(z) i (1 ΩDE) + (1 + z)ΩDE 1 αδ C(z) − − (1−ΩDE ) w (z) = 1 + , (7.43) DE 1−δ − Λ(1−ΩDE ) αδ n C(z) o (1 ΩDE) + 3 1 − C(z) − 2−δ (1−ΩDE )

0 where ΩDE and ΩDE are given by (7.40) and (7.42) respectively. Lastly, the decelaration parameter q 1 H˙ , using (7.38) is found to be ≡ − − H2

1 3Ω + 4Ω (1 + z) (1 + z)Ω0 q(z) = 1 + m0 r0 + DE . (7.44) − 2 Ωm0 + Ωr0(1 + z) (1 ΩDE) − In summary, in the case where radiation is present, we were able to extract analytical

solutions for ΩDE(z) and wDE(z), for the modiﬁed, nonextensive cosmological scenarios of the present work. CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 132

Cosmological evolution with Λ = 0

Let us now focus on the interesting case where the explicit cosmological constant is absent, namely when Λ = 0. This scenario was analyzed in subsection 7.3.2 above in the absence of radiation, however we now study it in the full case where radiation is included. For Λ = 0, relation (7.40) becomes

1 ( 3 4 δ−1 ) δ−2 [Ωm0(1 + z) + Ωr0(1 + z) ] ΩDE(z) = 1 , (7.45) − Ωm0 + Ωr0 relation (7.41) becomes

2 δ 2(δ−1) α = − H [Ωm0 + Ωr0] , (7.46) δ 0 and thus positivity of α implies that δ < 2, relation (7.42) becomes

1 3 4 δ−2 0 δ 1 [Ωm0(1 + z) + Ωr0(1 + z) ] ΩDE(z) = − −δ 2 [Ωm0 + Ωr0] − 2 3 3Ωm0(1 + z) + 4Ωr0(1 + z) , (7.47) ·

relation (7.43) becomes

3(1 δ)Ωm0 + (2 3δ)(1 + z)Ωr0 (δ 1)[3Ωm0 + 4(1 + z)Ωr0] wDE(z) = − − + − 3(δ 2)[Ωm0 + (1 + z)Ωr0] 3(δ 2)[Ωm0 + (1 + z)Ωr0] − 1 − , (7.48) ·n 3(1−δ) 1 δ−1 o 1 (1 + z) δ−2 (Ωm0 + Ωr0) δ−2 [Ωm0 + (1 + z)Ωr0] 2−δ − while relation (7.44) becomes

3 4 [Ωm0 + 2Ωr0(1 + z)] δ 1 [3Ωm0(1 + z) + 4Ωr0(1 + z) ] q = − 3 4 . (7.49) 2 [Ωm0 + Ωr0(1 + z)] − 2(δ 2) [Ωm0(1 + z) + Ωr0(1 + z) ] − We mention that relations (7.45)-(7.49) are the extensions of (7.29)-(7.33) in the presence of radiation. Let us examine this scenario in more detail, and in particular study the eﬀect of δ

on the cosmological evolution. In Fig. 7.7 we present wDE(z) for various choices of δ, extending the evolution up to the far future. In all cases the parameter α is set according CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 133

to (7.46) in order to obtain Ωm(z = 0) = Ωm0 = 0.3 and Ωr(z = 0) = Ωr0 = 0.000092 [361], and the expected thermal history of the universe. As we observe, for decreasing δ the wDE(z) tends to lower values. Moreover, although the asymptotic value of ΩDE(z) as z 1 is 1, as can be seen immediately from (7.45), namely the universe tends to the → − complete dark-energy domination, the asymptotic value of wDE is not the cosmological constant value 1, i.e the universe does not result in a de Sitter space. In particular, − from (7.48) we can see that for 1 δ < 2, wDE 0 as z 1, while for δ < 1, ≤ → → − wDE (δ 1)/(2 δ) as z 1. These asymptotic values are the same with the ones → − − → − in the absence of radiation mentioned in subsection 7.3.2, which was expected since at late times the eﬀect of radiation is negligible.

0.0 δ=1 δ=0.9 -0.5 δ=0.8 δ=0.7 -1.0

DE δ=0.6 w -1.5

-2.0

-2.5 δ=0.5 -3.0 -0.5 0.0 0.5 1.0 1.5 2.0 z

Figure 7.7: Evolution of wDE for Λ = 0 and for various values of the nonextensive parameter δ, in the case where radiation is present. For each value of δ we choose

α according to (7.46) in order to obtain Ωm(z = 0) = Ωm0 = 0.3 and Ωr(z = 0) =

Ωr0 = 0.000092 at present [361], and acquire the expected thermal history of the universe. Source: [354]

In summary, the scenario of modiﬁed cosmology through nonextensive thermodynamics, even in the case where an explicit cosmological constant is absent, is eﬃcient in describing the cosmological behavior of the universe. In order to present this behavior more transparently we confront the scenario with Supernovae type Ia (SN Ia) data. In CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 134 these observational sets the apparent luminosity l(z), or equivalently the apparent magnitude m(z), are measured as functions of the redshift, and are related to the luminosity distance as

L d (z) 2.5 log = µ m(z) M = 5 log L obs + 25, (7.50) l(z) ≡ − Mpc where M and L are the absolute magnitude and luminosity respectively. Additionally, for any theoretical model one can calculate the predicted dimensionless luminosity distance dL(z)th using the predicted evolution of the Hubble function as

Z z dz0 dL (z)th (1 + z) 0 . (7.51) ≡ 0 H (z )

Figure 7.8: The theoretically predicted apparent minus absolute magnitude as a function of the redshift, for the scenario of modiﬁed cosmology through nonextensive thermodynamics, for Λ = 0, in the case where radiation is present, for δ = 0.5 (red-dashed) and δ = 0.6 (green-dotted). The observational points correspond to the 580 SN Ia data points from [362], and for completeness and comparison we depict the prediction of ΛCDM cosmology with the black-solid curve. Source: [354]

In the scenario at hand, H(z) can be immediately calculated analytically from (7.37), knowing (7.39) and (7.40). In Fig. 7.8 we depict the theoretically predicted apparent minus absolute magnitude as a function of z, for two δ choices, as well as the prediction CHAPTER 7. MODIFIED COSMOLOGY THROUGH TSALLIS ENTROPY 135 of ΛCDM cosmology, on top of the 580 SN Ia observational data points from [362]. As we can see the agreement with the SN Ia data is excellent. The detailed comparison with observations, namely the joint analysis using data from SN Ia, Baryon Acoustic Oscillation (BAO), Cosmic Microwave Background (CMB), and direct Hubble parameter observations, lies beyond the scope of the present work and it is left for a future project. We close this subsection mentioning that the present scenario is very efficient in mim- icking the cosmological constant, despite the fact that in this case the exact ΛCDM cosmology cannot be obtained for any parameter values. In particular, choosing the nonextensive parameter δ suitably (namely δ 0.5 0.6) we obtain agreement with ∼ − observations. This is a significant result that shows the capabilities of the modified cosmology through nonextensive thermodynamics. Chapter 8

Inﬂation using non-canonical scalars

In this work [104] we revisit inﬂation with non-canonical scalar ﬁelds by applying deformed- steepness exponential potentials. Non-canonical kinetic terms can arise naturally in models of supergravity and superstrings, while exponential potentials have remarkable properties, as they greatly facilitate slow roll and result in scaling behaviour at large scales.

8.1 Non-canonical inﬂation with deformed-steepness potentials

The usual non-canonical Lagrangian, which is well justified theoretically, takes the form [40,66,67,72,119,363], X α−1 (φ, X) = X V (φ), (8.1) L M 4 − 1 µ where X = 2 ∂µφ∂ φ is the kinetic energy of the scalar field, and thus the action of the scenario reads " # Z R X α−1 = d4x√ g M 2 + X V (φ) . (8.2) S − pl 2 M 4 − The parameter M has dimensions of mass and determines the scale in which the non- canonical effects become significant, while Mpl is the Planck mass. Concerning the potential, in this work we will consider the deformed-steepness potential that was introduced in [96,98], namely −λφn/M n V (φ) = V0 e pl , (8.3)

136 CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 137

with V0 and λ the usual potential parameters and n the new exponent parameter that determines the deformed-steepness. We consider a homogeneous and isotropic ﬂat Friedmann-Robertson-Walker (FRW) metric of the form

2 2 2 i j ds = dt + a (t)δijdx dx , (8.4) − where a(t) is the scale factor. Variation of the action (8.2) in terms of the metric gives the following Friedmann equations

" α−1 # 1 X n n 2 −λφ /Mpl H = 2 (2α 1)X 4 + V0 e (8.5) 3Mpl − M

α−1 ˙ 1 X H = 2 αX 4 , (8.6) −Mpl M a˙ where H = a is the Hubble parameter. Additionally, variation in terms of the scalar ﬁeld leads to the Klein-Gordon equation

−λφn/M n α−1 3Hφ˙ λnφn−1V e pl 2M 4 φ¨ + 0 = 0. (8.7) 2α 1 − α(2α 1)M n φ˙2 − − pl Note that one can write the above equation in the form of the usual conservation equation

ρ˙φ + 3H(ρφ + pφ), using the deﬁnitions α−1 X −λφn/M n ρφ = (2α 1)X + V0 e pl − M 4

α−1 X −λφn/M n pφ = X V0 e pl . (8.8) M 4 − In every inflationary scenario the important quantities are the inflation-related observables, namely the scalar spectral index of the curvature perturbations ns and its ~ running αs dns/d ln k, with k the measure of the wave number k, the tensor spectral ≡ index nT and its running, as well as the tensor-to-scalar ratio r. In a given scenario these quantities depend on the model parameters, and hence confrontation with observational data can lead to constraints on these model parameters. In order to extract the relations for the inflation-related observables, a detailed and thorough perturbation analysis is needed. In the simple case of canonical fields minimally coupled to gravity, and introducing the slow-roll parameters, full perturbation analysis indicates that the inflationary observables can be expressed solely in terms of the scalar CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 138

potential and its derivatives [26,28,35]. However, in the case where non-canonical terms or forms of non-minimally coupling are present, as well as in the case where the potential itself is absent (as for instance in modiﬁed gravity inﬂation), one should instead introduce

the Hubble slow-roll parameters n (with n positive integer), deﬁned as [28,364–366]

d ln n n+1 | |, (8.9) ≡ dN where N ln(a/aini) is the e-folding number, and 0 Hini/H, where aini is the initial ≡ ≡ scale factor with Hini the corresponding Hubble parameter (as usual inﬂation ends when

1 = 1). Thus, the ﬁrst three n are found to be H˙ 1 , (8.10) ≡ −H2 H¨ 2H˙ 2 , (8.11) ≡ HH˙ − H2 " ... # −1 ˙ ¨ ˙ 2 ¨ ˙ ¨ ˙ 2 HHH H(H + HH) 2H ¨ ˙ 2 3 HH 2H − (HH 2H ) . (8.12) ≡ − HH˙ − H2 −

With these deﬁnitions, the basic inﬂationary observables are given as [28]

r 16cs1, (8.13) ≈

ns 1 21 2, (8.14) ≈ − −

αs 212 23, (8.15) ≈ − −

nT 21, (8.16) ≈ − where the sound speed is defined as [72] (∂ /∂X) c2 = L , (8.17) s (∂ /∂X) + (2X)(∂2 /∂X2) L L which for the Lagrangian (8.1) gives 1 c2 = . (8.18) s 2α 1 − In the scenario of non-canonical inflation with deformed-steepness potentials, described by equations (8.5)-(8.7), the dynamics, i.e. the Hubble function, is determined by the parameters α and M related to “non-canonicality”, by the standard potential parameters V0 and λ, alongside the deformed-steepness parameter n. Hence, we deduce that the above inflationary observables (8.13)-(8.16) will be determined by these model parameters too. Nevertheless, we should mention that these parameters are not independent CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 139 since they are related through the observed value of the amplitude of scalar perturbation

S. In particular, for the action (8.2) one ﬁnds [72] A 1 α 5α−2 2α−1 1 −2(5α−6) α6 V (φ) S = 2 Mpl 4(α−1) 0 2α , (8.19) P 72π cs M V (φ) which for our potential (8.3) leads to

1 " φn 2α(1−n)# 2α−1 −(3α−2)λ Mn φ S = S e pl , (8.20) P A Mpl with 1 α 3α−2 2α−1 1 4(1−2α) α6 V0 S = 2 Mpl 4(α−1) 2α . (8.21) A 72π cs M (λn) Since we know that Planck observations give [42]

−9 Sobs 2.09052 10 , (8.22) A ' · in the following analysis we set V0 at will while λ arises from the satisfaction of the above observational constraint. In the next section we investigate in detail the eﬀect of each parameter on the inﬂa- tionary observables, and we will show which combinations can bring the predictions deep inside the observational contours.

8.2 Results

In this section, we investigate the inflationary observables in the scenario of non-canonical inflation with deformed-steepness potentials. In particular, we desire to see how the scalar spectral index ns and the tensor-to-scalar ratio r are affected by the model parameters. Since the involved equations (8.5)-(8.7), the slow-roll parameters (8.10)-(8.12) and the observables expressions (8.13)-(8.16) are in general too complicated to admit analytical solutions, we investigate them numerically. Specifically, for a given set of parameter values we impose the conditions for φ, φ˙ and H corresponding to small i. We evolve the system and we determine the end of inflation by demanding 1 = 1 (cases of eternal inflation are considered non-physical), and thus by imposing the desired e-folding number N we extract the time at the beginning of inflation. Hence, we can use the corresponding Hubble parameter to calculate the CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 140

α = 2, n = 3, λ = 7.54 10−6 α = 2, n = 4, λ = 5.65 10−6 α = 2, n = 5, λ = 4.52 10−6 · · · N 50 60 70 N 50 60 70 N 50 60 70 r 0.0445 0.0359 0.0299 r 0.0341 0.0263 0.0209 r 0.0240 0.0174 0.0130 ns 0.9670 0.9723 0.9761 ns 0.9649 0.9700 0.9736 ns 0.9611 0.9659 0.9693

α = 4, n = 3, λ = 7.61 10−6 α = 4, n = 4, λ = 5.71 10−6 α = 4, n = 5, λ = 4.57 10−6 · · · N 50 60 70 N 50 60 70 N 50 60 70 r 0.0350 0.0288 0.0244 r 0.0318 0.0257 0.0214 r 0.0282 0.0224 0.0183 ns 0.9670 0.9724 0.9763 ns 0.9666 0.9719 0.9758 ns 0.9657 0.9710 0.9748

α = 6, n = 3, λ = 8.25 10−6 α = 6, n = 4, λ = 6.19 10−6 α = 6, n = 5, λ = 4.95 10−6 · · · N 50 60 70 N 50 60 70 N 50 60 70 r 0.0289 0.0238 0.0202 r 0.0268 0.0218 0.0183 r 0.0245 0.0197 0.0163 ns 0.9668 0.9724 0.9762 ns 0.9660 0.9700 0.9759 ns 0.9653 0.9714 0.9746

Table 8.1: Predictions for the scalar spectral index ns and the tensor-to-scalar ratio r for the scenario of non-canonical inﬂation with deformed-steepness potential, for various combinations of α and n, adjusting the values of λ in order to satisfy the observational constraint (8.22), and for e-folding number N equal to 50, 60 and 70. For this Table we

−4 −16 4 18 ﬁx M = 10 Mpl and V0 = 10 (GeV ) , with Mpl = 10 GeV , while the value of λ is determined through the observational constraint (8.22).

inﬂationary observables corresponding to the given parameter values and the imposed e-folding number N. We start our investigation by examining the eﬀect of the non-canonical parameter α

and the deformed-steepness parameter n. Therefore, we ﬁx M and V0 at theoretically

motivated values and we calculate ns and r for various combinations of α and n, adjusting suitably only the value of λ in order to satisfy the observational constraint (8.22), and for the e-folding number N taking, as usual, the values 50, 60 and 70. In Table 8.1 we summarise the obtained observable predictions. Additionally, in order to present the information in a more transparent way that allows comparison with observational data, in Fig. 8.1 we depict the results of Table 8.1 on top of the 1σ and 2σ contours of the Planck 2018 data [42]. A general observation is that the predictions of the scenario at hand lie well inside CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 141

0.15 0.15

r r

0.10 0.10

0.05 0.05

0.00 0.00

0.95 0.96 0.97 0.98 0.95 0.96 0.97 0.98

0.15

0.10

0.05

0.00

0.95 0.96 0.97 0.98

Figure 8.1: 1σ (yellow) and 2σ (light yellow) contours for Planck 2018 results (Planck

+TT + lowP ) [42], on the ns r plane. Furthermore, we depict the predictions of Table − 8.1, for three diﬀerent values of the the non-canonical parameter α and the deformed-

−6 steepness parameter n respectively. In this case, we keep fixed M = 10 Mpl and V0 = −16 4 18 10 (GeV ) , with Mpl = 10 GeV. In every line the black point corresponds to e-folding number N = 50, the red point to N = 60 and the green to N = 70. Source: [104] the 1σ region of the Planck 2018 data, without the need to use large values for the non-canonical parameter α or the deformed-steepness parameter n, which was indeed the main motivation behind the present work. Additionally, the predictions of the scenario are better, compared to the simple non-canonical models, as well as to the simple deformed- steepness models. Concerning the specific features, we find the following: For any given set of model- parameters, increasing the e-folding values N leads to increased ns and decreased r, as CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 142

is usual in the majority of inﬂationary scenarios. Now, for a given α, as n increases

both ns and r also increase. On the other hand, for a given n, as α increases there is

no particular tendency for ns and r. However, for larger α, such as α = 6, the effect of n is less significant and the different curves actually coincide. Moreover, as α grows, λ slightly increases, while as n grows λ slightly decreases.

−6 α = 2, n = 4, M = 10 Mpl −16 4 −17 4 −18 4 V0 10 (GeV ) 10 (GeV ) 10 (GeV ) λ 5.65 10−6 5.65 10−7 5.65 10−8 · · · N 50 60 70 50 60 70 50 60 70 r 0.0341 0.0263 0.0209 0.0475 0.0387 0.0325 0.0540 0.0447 0.0382

ns 0.9649 0.9700 0.9736 0.9674 0.9728 0.9766 0.9678 0.9732 0.9769

−16 4 α = 2, n = 4, V0 = 10 (GeV ) M 10−6 5 10−7 10−7 · λ 5.65 10−6 1.13 10−5 5.65 10−5 · · · N 50 60 70 50 60 70 50 60 70 r 0.0341 0.0263 0.0209 0.0415 0.0332 0.0273 0.0504 0.0414 0.0351

ns 0.9649 0.9700 0.9736 0.9667 0.9720 0.9757 0.9678 0.9731 0.9769

Table 8.2: Predictions for the scalar spectral index ns and the tensor-to-scalar ratio r for the scenario of non-canonical inﬂation with deformed-steepness potential, for ﬁxed α = 2,

n = 4, for e-folding number N being 50, 60 and 70, and with ﬁxed M and varying V0

(upper sub-Table) and with ﬁxed V0 and varying M (lower sub-Table). In all cases the 18 value of λ is determined through the observational constraint (8.22), while Mpl = 10 GeV .

We proceed by investigating the effect on the observables of the parameters M and V0, which determine the scale of non-canonicality and of the potential, respectively, keeping in mind that the scale of inflation in theoretically motivated constructions can be anywhere from just below the unification scale (mostly Grand Unification), to energies as low as the scales within reach of the LHC (see e.g. [367]), with many possibilities in between, that can be linked i.e. to different stages of symmetry breaking. Without loss of generality

we ﬁx α = 2 and n = 4 and we calculate ns and r for e-folding number N being as usual CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 143

50, 60 and 70. We first additionally fix M and change V0, and then we fix V0 and change M. In all cases we adjust the value of λ in order to satisfy the observational constraint (8.22). We summarise the obtained observable predictions in Table 8.2. Moreover, in order to present the results in a more transparent way, in Fig. 8.2 we depict the results of Table 8.2 on top of the 1σ and 2σ contours of the Planck 2018 data [42].

0.15 -6 0.15 -16

M=10 M V =10 (GeV)

pl 0

0.10 0.10

0.05 0.05

0.00 0.00

0.95 0.96 0.97 0.98 0.95 0.96 0.97 0.98

Figure 8.2: 1σ (yellow) and 2σ (light yellow) contours for Planck 2018 results (Planck

+TT + lowP ) [42], on the ns r plane. Moreover, we depict the predictions of Table 8.2, − for ﬁxed α = 2, n = 4. As in Fig. 8.1 the black point corresponds to e-folding number

−6 N = 50, red point to N = 60 and the green to N = 70. Left panel: Fixed M = 10 Mpl. −16 4 −17 4 Black - solid for V0 = 10 (GeV ) , blue - dashed for V0 = 10 (GeV ) , green - dotted −18 4 −16 4 for V0 = 10 (GeV ) . Right panel: Fixed V0 = 10 (GeV ) . Black - solid for −6 −7 −7 M = 10 Mpl, blue - dashed for M = 5 10 Mpl, green - dotted for M = 10 Mpl. In · all cases the value of λ is determined through the observational constraint (8.22), while

18 Mpl = 10 GeV . Source: [104]

The main observation is that the predictions of the scenario lie well inside the 1σ region of observational data. Now, for ﬁxed M, increasing V0 leads to lower values of r and ns; moreover, the variation of r is much faster than that of ns. Additionally, for ﬁxed

V0, increasing M leads also to lower values of r and ns; nevertheless the change in ns is strongly aﬀected by the change in M (since M appears in powers of four in the equations) so that it can easily be led outside the observational contours. This similar tendency CHAPTER 8. INFLATION USING NON-CANONICAL SCALARS 144

behavior was expected, since in the scalar-ﬁeld equation (8.7) the two parameters M and

V0 appear multiplied. Nevertheless, this is not a trivial result, since M is related to the non-canonicality scale while V0 to the potential scale. From the above analysis we deduce that non-canonical kinetic terms combined with deformed-steepness potentials can provide inﬂationary predictions in very good agreement with observations, compared to simple non-canonical models [68–85] as well as to canonical models with deformed-steepness potentials [96–103]. An additional signiﬁcant advantage is that the above combination allows good predictions without the need to use unnaturally large values for α or n, or unnaturally tuned values for the non-canonicality and potential scales M and V0, as well as for the potential exponent λ. In particular, we

see that M and V0 remain in reasonable sub-Planckian regions, with values that can be easily predicted and accepted from a field theoretical point of view. This combination of observational efficiency and theoretical justification is a significant advantage of the scenario at hand. Chapter 9

Discussion-Conclusions

In chapter 6 we have derived analytically and numerically the cosmological solution close to a future-time singularity for both quintessence and scalar-tensor quintessence models. ... For quintessence, we have shown that there is a divergence of a and a GSFS singularity occurs (as, ρs, ps remain finite butp ˙ ) , while in the case of scalar-tensor quintessence → ∞ models there is a divergence ofa ¨ and an SFS singularity occurs (as, ρs remain finite but ps ,p ˙ ). Introducing a perfect fluid in the dynamical equations, in both cases, → ∞ → ∞ we have shown that this result is still valid in our cosmological solution. These are the simplest non-exotic physical models where GSFS and SFS singularities naturally arise. In the case of scalar-tensor quintessence models, there is a divergence

2 of the scalar curvature R = 6 a¨ + a˙ because of the divergence of the second a a2 → ∞ derivative of the scale factor. Thus, a stronger singularity occurs in this class of models. Such divergence of the scalar curvature is not present in the simple quintessence case. We have also shown the important role of the additional linear and quadratic terms

of ts t in the form of the scale factor as t ts. However, in the scalar-tensor case the − → quadratic term becomes subdominant close to the singularity. We have derived explicitly the relations between the coefficients of the linear, quadratic and diverging terms of the scale factor and the scalar field. We have shown that all coefficients of the fluid case (quintessence and scalar-tensor quintessence), reduce to those

of the no fluid case for ρ0m = 0, and all coefficients (except coefficient d) of the scalar- tensor models reduce to those of the simple quintessence, in the special case λ = 0 i.e. F = 1. Moreover, for quintessence models, we derived relations for the Hubble parameter,

145 CHAPTER 9. DISCUSSION-CONCLUSIONS 146

˙ 2 ˙ 3 3 2 H = 3H (for the no fluid case) and H = Ω0,m(1 + zs) 3H (for the fluid case), − 2 − close to the singularity. These relations may be used as observational signatures of such singularities in this class of models. Interesting extensions of this analysis include the study of the strength of these singularities in other modified gravity models e.g. string-inspired gravity, Gauss-Bonnet gravity etc [260, 368] and the search for signatures of such singularities in cosmological luminosity distance and angular diameter distance data. In chapter 7 we constructed a modified cosmological scenario through the application of the first law of thermodynamics, but using the generalized, nonextensive Tsallis entropy instead of the usual Bekenstein-Hawking one. In particular, there is a well-studied procedure in the literature, which works for a variety of modified gravities, where one can apply the first law of thermodynamics in the universe horizon and extract the Friedmann equations. The crucial part in this procedure is the use of the modified entropy relation of the specific modified gravity, which is known only after this modified gravity is given, and thus in this sense it cannot provide new gravitational modifications. However, if we apply this approach using the nonextensive, Tsallis entropy, which is the consistent concept that should be used in non-additive gravitational systems such as the whole universe, then we obtain modified cosmological equations that possess the usual ones as a particular limit, but which in the general case contains terms that appear for the first time. The new terms that appear in the modified Friedmann equations are quantified by the nonextensive parameter δ and constitute an effective dark energy sector. In the case where Tsallis entropy becomes the usual Bekenstein-Hawking entropy, namely when δ = 1, the effective dark energy coincides with the cosmological constant and ΛCDM cosmology is restored. However, in the general case the scenario of modified cosmology at hand presents very interesting cosmological behavior. When the matter sector is dust, we were able to extract analytical expressions for the dark energy density and equation-of-state parameters, and we extended these solutions in the case where radiation is present too. These solutions show that the universe exhibits the usual thermal history, with the sequence of matter and dark-energy eras and the onset of acceleration at around z 0.5 in agreement with observations. In the case ≈ CHAPTER 9. DISCUSSION-CONCLUSIONS 147 where an explicit cosmological constant is present, depending on the value of δ the dark- energy equation-of-state parameter exhibits a very interesting behavior and it can be quintessence-like, phantom-like, or experience the phantom-divide crossing during the evolution, before it asymptotically stabilizes in the cosmological constant value 1 in the − far future. An interesting sub-case of the scenario of modified cosmology through nonextensive thermodynamics is when we set the explicit cosmological constant to zero, since in this case the universe evolution is driven solely by the news terms. Extracting analytical solutions for the dark energy density and equation-of-state parameters we showed that indeed the new terms can very efficiently mimic ΛCDM cosmology, even though Λ is absent, with the successive sequence of matter and dark energy epochs, before the universe results in complete dark-energy domination in the far future. Moreover, confronting the model with SN Ia data we saw that the agreement is excellent. In summary, modified cosmology through nonextensive thermodynamics is very efficient in describing the universe evolution, and thus it can be a candidate for the description of nature. In the present work we derived the cosmological equations by applying the well-known thermodynamics procedure to the universe horizon. It would be interesting to investigate whether these equations can arise from a nonextensive action too. Such a study is left for a future project. In chapter 8 we revisited inflation with non-canonical scalar fields by applying deformed- steepness exponential potentials. Non-canonical kinetic terms can arise naturally in models of supergravity and superstrings, while exponential potentials have remarkable properties, as they greatly facilitate slow roll and result to scaling behaviour at large scales. As we have shown, the resulting scenario can lead to inflationary observables, and in particular to scalar spectral index of the curvature perturbations ns and tensor-to- scalar ratio r, in remarkable agreement with the observations of Planck 2018, being well inside the 1σ region. Apart from observational predictability, a significant additional advantage of the proposed scenario arises from a theoretical point of view. In particular, in order to obtain acceptable observables, in simple non-canonical models one needs to use relatively large non-canonical exponent α or ranges of values for the non-canonicality scale M, while in canonical models with deformed-steepness potentials relatively large CHAPTER 9. DISCUSSION-CONCLUSIONS 148 values of the extra exponent n need to be imposed, and, hence, these models cannot be well-justified theoretically. On the other hand, in the scenario of the present work the exponents α and n are small, as well as the non-canonicality and potential scales M and

V0 remaining in reasonable sub-Planckian regions.

Our analysis revealed that, for a given α, as n increases both ns and r decrease too, while on the other hand, for a given n, as α increases there is no particular tendency for ns and r. Additionally, for fixed M, increasing V0 leads to lower values of r and ns, while for fixed V0, increasing M leads to lower values of r and ns too. In summary, we showed that revisiting non-canonical inflation models by applying potentials with deformed steepness, increases the observational predictability, bringing the scalar spectral index and the tensor-to-scalar ratio more deeply into the observational contours, offering a better theoretical justification for the required parameters. This combination of observational efficiency and theoretical justification is a significant advantage of the scenario at hand, and hence, non-canonical models with deformed-steepness potential need to be further explored, as additional observational data will be coming forward. Bibliography

[1] Albert Einstein. The Foundation of the General Theory of Relativity. Annalen Phys., 49(7):769–822, 1916.

[2] N. Aghanim et al. Planck 2018 results. VI. Cosmological parameters. 7 2018.

[3] Arno A. Penzias and Robert Woodrow Wilson. A Measurement of excess antenna temperature at 4080-Mc/s. Astrophys. J., 142:419–421, 1965.

[4] G. Efstathiou, J.R. Bond, and Simon D.M. White. COBE Background radiation anisotropies and large scale structure in the universe. Mon. Not. Roy. Astron. Soc., 258:1–6, 1992.

[5] Lev A. Kofman, Nickolay Y. Gnedin, and Neta A. Bahcall. Cosmological constant, COBE cosmic microwave background anisotropy, and large scale clustering. Astrophys. J., 413:1–9, 1993.

[6] V. Mukhanov. Physical Foundations of Cosmology. Cambridge University Press, Oxford, 2005.

[7] Steven Weinberg. The First Three Minutes. A Modern View of the Origin of the Universe. 1 1977.

[8] Adam G. Riess et al. Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J., 116:1009–1038, 1998.

[9] S. Perlmutter et al. Measurements of Ω and Λ from 42 high redshift supernovae. Astrophys. J., 517:565–586, 1999.

[10] Edwin Hubble. A relation between distance and radial velocity among extra-galactic nebulae. Proc. Nat. Acad. Sci., 15:168–173, 1929.

149 BIBLIOGRAPHY 150

[11] A. Friedman. On the Curvature of space. Z. Phys., 10:377–386, 1922.

[12] A. Friedmann. On the Possibility of a world with constant negative curvature of space. Z. Phys., 21:326–332, 1924.

[13] H.P. Robertson. Relativistic Cosmology. Rev. Mod. Phys., 5:62–90, 1933.

[14] H.P. Robertson. Kinematics and World-Structure. Astrophys. J., 82:284–301, 1935.

[15] H.P. Robertson. Kinematics and World-Structure. 2. Astrophys. J., 83:187–201, 1935.

[16] H.P. Robertson. Kinematics and World-Structure. 3. Astrophys. J., 83:257–271, 1936.

[17] G. Lemaitre. The expanding universe. Gen. Rel. Grav., 29:641–680, 1997.

[18] Alan H. Guth. The Inﬂationary Universe: A Possible Solution to the Horizon and Flatness Problems. Adv. Ser. Astrophys. Cosmol., 3:139–148, 1987.

[19] Andrei D. Linde. A New Inﬂationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems. Adv. Ser. Astrophys. Cosmol., 3:149–153, 1987.

[20] S.W. Hawking. The Development of Irregularities in a Single Bubble Inﬂationary Universe. Phys. Lett. B, 115:295, 1982.

[21] Alexei A. Starobinsky. Dynamics of Phase Transition in the New Inﬂationary Uni- verse Scenario and Generation of Perturbations. Phys. Lett. B, 117:175–178, 1982.

[22] Alexander Vilenkin. Quantum Fluctuations in the New Inﬂationary Universe. Nucl. Phys. B, 226:527–546, 1983.

[23] Andrei D. Linde. Scalar Field Fluctuations in Expanding Universe and the New Inﬂationary Universe Scenario. Phys. Lett. B, 116:335–339, 1982.

[24] Alan H. Guth and S.Y. Pi. Fluctuations in the New Inﬂationary Universe. Phys. Rev. Lett., 49:1110–1113, 1982. BIBLIOGRAPHY 151

[25] Andrei D. Linde. Chaotic Inﬂation. Phys. Lett. B, 129:177–181, 1983.

[26] David H. Lyth and Antonio Riotto. Particle physics models of inﬂation and the cosmological density perturbation. Phys. Rept., 314:1–146, 1999.

[27] Andrei D. Linde. Inﬂationary Cosmology. Lect. Notes Phys., 738:1–54, 2008.

[28] Jerome Martin, Christophe Ringeval, and Vincent Vennin. Encyclopaedia Inﬂa- tionaris. Phys. Dark Univ., 5-6:75–235, 2014.

[29] Andrew R. Liddle, Paul Parsons, and John D. Barrow. Formalizing the slow roll approximation in inﬂation. Phys. Rev. D, 50:7222–7232, 1994.

[30] Lev Kofman, Andrei D. Linde, and Alexei A. Starobinsky. Reheating after inﬂation. Phys. Rev. Lett., 73:3195–3198, 1994.

[31] Lev Kofman, Andrei D. Linde, and Alexei A. Starobinsky. Towards the theory of reheating after inﬂation. Phys. Rev. D, 56:3258–3295, 1997.

[32] Bruce A. Bassett, Shinji Tsujikawa, and David Wands. Inﬂation dynamics and reheating. Rev. Mod. Phys., 78:537–589, 2006.

[33] Andrew R Liddle and Samuel M Leach. How long before the end of inﬂation were observable perturbations produced? Phys. Rev. D, 68:103503, 2003.

[34] Scott Dodelson and Lam Hui. A Horizon ratio bound for inﬂationary ﬂuctuations. Phys. Rev. Lett., 91:131301, 2003.

[35] James E. Lidsey, Andrew R. Liddle, Edward W. Kolb, Edmund J. Copeland, Tiago Barreiro, and Mark Abney. Reconstructing the inﬂation potential : An overview. Rev. Mod. Phys., 69:373–410, 1997.

[36] James M. Bardeen. Gauge Invariant Cosmological Perturbations. Phys. Rev. D, 22:1882–1905, 1980.

[37] Viatcheslav F. Mukhanov, H.A. Feldman, and Robert H. Brandenberger. Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions. Phys. Rept., 215:203–333, 1992. BIBLIOGRAPHY 152

[38] J.M. Stewart. Perturbations of Friedmann-Robertson-Walker cosmological models. Class. Quant. Grav., 7:1169–1180, 1990.

[39] Hideo Kodama and Misao Sasaki. Cosmological Perturbation Theory. Prog. Theor. Phys. Suppl., 78:1–166, 1984.

[40] Jaume Garriga and Viatcheslav F. Mukhanov. Perturbations in k-inﬂation. Phys. Lett. B, 458:219–225, 1999.

[41] Misao Sasaki. Large Scale Quantum Fluctuations in the Inﬂationary Universe. Prog. Theor. Phys., 76:1036, 1986.

[42] Y. Akrami et al. Planck 2018 results. X. Constraints on inﬂation. 7 2018.

[43] Alexei A. Starobinsky. A New Type of Isotropic Cosmological Models Without Singularity. Phys. Lett., B91:99–102, 1980. [,771(1980)].

[44] D. Kazanas. Dynamics of the Universe and Spontaneous Symmetry Breaking. As- trophys. J. Lett., 241:L59–L63, 1980.

[45] K. Sato. First Order Phase Transition of a Vacuum and Expansion of the Universe. Mon. Not. Roy. Astron. Soc., 195:467–479, 1981.

[46] Keith A. Olive. Inﬂation. Phys. Rept., 190:307–403, 1990.

[47] N. Bartolo, E. Komatsu, Sabino Matarrese, and A. Riotto. Non-Gaussianity from inﬂation: Theory and observations. Phys. Rept., 402:103–266, 2004.

[48] Shin’ichi Nojiri and Sergei D. Odintsov. Uniﬁed cosmic history in modiﬁed gravity: from F(R) theory to Lorentz non-invariant models. Phys. Rept., 505:59–144, 2011.

[49] Shin’ichi Nojiri and Sergei D. Odintsov. Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration. Phys. Rev., D68:123512, 2003.

[50] Benedict M.N. Carter and Ishwaree P. Neupane. Towards inﬂation and dark energy cosmologies from modiﬁed Gauss-Bonnet theory. JCAP, 06:004, 2006. BIBLIOGRAPHY 153

[51] Rafael Ferraro and Franco Fiorini. Modified teleparallel gravity: Inflation without inflaton. Phys. Rev. D, 75:084031, 2007.

[52] Cristiano Germani and Alex Kehagias. New Model of Inﬂation with Non-minimal Derivative Coupling of Standard Model Higgs Boson to Gravity. Phys. Rev. Lett., 105:011302, 2010.

[53] L. Sebastiani, G. Cognola, R. Myrzakulov, S.D. Odintsov, and S. Zerbini. Nearly Starobinsky inﬂation from modiﬁed gravity. Phys. Rev. D, 89(2):023518, 2014.

[54] John R. Ellis, Dimitri V. Nanopoulos, Keith A. Olive, and K. Tamvakis. Cosmo- logical Inﬂation Cries Out for Supersymmetry. Phys. Lett. B, 118:335, 1982.

[55] John R. Ellis, Andrei D. Linde, and Dimitri V. Nanopoulos. Inﬂation Can Save the Gravitino. Phys. Lett. B, 118:59–64, 1982.

[56] G.R. Dvali, Q. Shafi, and Robert K. Schaefer. Large scale structure and supersymmetric inflation without fine tuning. Phys. Rev. Lett., 73:1886–1889, 1994.

[57] Graham G. Ross and Subir Sarkar. Successful supersymmetric inﬂation. Nucl. Phys. B, 461:597–624, 1996.

[58] Dimitri V. Nanopoulos, Keith A. Olive, M. Srednicki, and K. Tamvakis. Primordial Inﬂation in Simple Supergravity. Phys. Lett. B, 123:41–44, 1983.

[59] Ewan D. Stewart. Inﬂation, supergravity and superstrings. Phys. Rev. D, 51:6847– 6853, 1995.

[60] Edi Halyo. Hybrid inﬂation from supergravity D terms. Phys. Lett. B, 387:43–47, 1996.

[61] Andrei D. Linde and Antonio Riotto. Hybrid inﬂation in supergravity. Phys. Rev. D, 56:1841–1844, 1997.

[62] Q. Shaﬁ and C. Wetterich. Inﬂation From Higher Dimensions. Nucl. Phys. B, 289:787–809, 1987.

[63] Ignatios Antoniadis, C. Bachas, John R. Ellis, and Dimitri V. Nanopoulos. Cosmo- logical String Theories and Discrete Inﬂation. Phys. Lett. B, 211:393–399, 1988. BIBLIOGRAPHY 154

[64] G.R. Dvali and S.H.Henry Tye. Brane inﬂation. Phys. Lett. B, 450:72–82, 1999.

[65] Fred C. Adams, J.Richard Bond, Katherine Freese, Joshua A. Frieman, and An- gela V. Olinto. Natural inﬂation: Particle physics models, power law spectra for large scale structure, and constraints from COBE. Phys. Rev. D, 47:426–455, 1993.

[66] C. Armendariz-Picon, T. Damour, and Viatcheslav F. Mukhanov. k - inﬂation. Phys. Lett. B, 458:209–218, 1999.

[67] Viatcheslav F. Mukhanov and Alexander Vikman. Enhancing the tensor-to-scalar ratio in simple inﬂation. JCAP, 02:004, 2006.

[68] Gabriela Barenboim and William H. Kinney. Slow roll in simple non-canonical inﬂation. JCAP, 03:014, 2007.

[69] Konstantinos Tzirakis and William H. Kinney. Non-canonical generalizations of slow-roll inﬂation models. JCAP, 01:028, 2009.

[70] Paul Franche, Rhiannon Gwyn, Bret Underwood, and Alisha Wissanji. Attractive Lagrangians for Non-Canonical Inﬂation. Phys. Rev. D, 81:123526, 2010.

[71] Paul Franche, Rhiannon Gwyn, Bret Underwood, and Alisha Wissanji. Initial Conditions for Non-Canonical Inﬂation. Phys. Rev. D, 82:063528, 2010.

[72] Sanil Unnikrishnan, Varun Sahni, and Aleksey Toporensky. Reﬁning inﬂation using non-canonical scalars. JCAP, 08:018, 2012.

[73] Rhiannon Gwyn, Markus Rummel, and Alexander Westphal. Relations between canonical and non-canonical inﬂation. JCAP, 12:010, 2013.

[74] Damien A. Easson and Brian A. Powell. The Degeneracy Problem in Non-Canonical Inﬂation. JCAP, 03:028, 2013.

[75] Xiao-Min Zhang and jian-Yang Zhu. Extension of warm inﬂation to noncanonical scalar ﬁelds. Phys. Rev. D, 90(12):123519, 2014.

[76] Rhiannon Gwyn and Jean-Luc Lehners. Non-Canonical Inﬂation in Supergravity. JHEP, 05:050, 2014. BIBLIOGRAPHY 155

[77] Md. Wali Hossain, R. Myrzakulov, M. Sami, and Emmanuel N. Saridakis. Vari- able gravity: A suitable framework for quintessential inﬂation. Phys. Rev. D, 90(2):023512, 2014.

[78] K. Rezazadeh, K. Karami, and P. Karimi. Intermediate inﬂation from a non- canonical scalar ﬁeld. JCAP, 09:053, 2015.

[79] Sebasti Cspedes and Anne-Christine Davis. Non-canonical inﬂation coupled to matter. JCAP, 11:014, 2015.

[80] Nina K. Stein and William H. Kinney. Planck Limits on Non-canonical General- izations of Large-ﬁeld Inﬂation Models. JCAP, 04:006, 2017.

[81] Konstantinos Dimopoulos and Charlotte Owen. Quintessential Inﬂation with α- attractors. JCAP, 06:027, 2017.

[82] Abolhassan Mohammadi, Khaled Saaidi, and Haidar Sheikhahmadi. Constant-roll approach to non-canonical inﬂation. Phys. Rev. D, 100(8):083520, 2019.

[83] Mahdeyeh Naderi, Ali Aghamohammadi, Abdollah Refaei, and Haidar Sheikhah- madi. Intermediate inﬂation with non-canonical scalar ﬁeld in the low anisotropy Universe. Mod. Phys. Lett. A, 34(33):1950272, 2019.

[84] Alexander Y. Kamenshchik, Alessandro Tronconi, Tereza Vardanyan, and Giovanni Venturi. Non-Canonical Inﬂation and Primordial Black Holes Production. Phys. Lett. B, 791:201–205, 2019.

[85] Tuan Q. Do, W.F. Kao, and Ing-Chen Lin. CMB imprints of non-canonical anisotropic inﬂation. 3 2020.

[86] Nilok Bose and A.S. Majumdar. A k-essence Model Of Inﬂation, Dark Matter and Dark Energy. Phys. Rev. D, 79:103517, 2009.

[87] Josue De-Santiago and Jorge L. Cervantes-Cota. Generalizing a Uniﬁed Model of Dark Matter, Dark Energy, and Inﬂation with Non Canonical Kinetic Term. Phys. Rev. D, 83:063502, 2011. BIBLIOGRAPHY 156

[88] Varun Sahni and Anjan A. Sen. A new recipe for Λ CDM. Eur. Phys. J. C, 77(4):225, 2017.

[89] Swagat S. Mishra and Varun Sahni. Unifying Dark Matter and Dark Energy with non-Canonical Scalars. 3 2018.

[90] Tsutomu Kobayashi, Masahide Yamaguchi, and Jun’ichi Yokoyama. G-inflation: Inflation driven by the Galileon field. Phys. Rev. Lett., 105:231302, 2010.

[91] Clare Burrage, Claudia de Rham, David Seery, and Andrew J. Tolley. Galileon inﬂation. JCAP, 01:014, 2011.

[92] Junko Ohashi and Shinji Tsujikawa. Potential-driven Galileon inﬂation. JCAP, 10:035, 2012.

[93] Shinji Tsujikawa. The effective field theory of inflation/dark energy and the Horn- deski theory. Lect. Notes Phys., 892:97–136, 2015.

[94] Lorenzo Sebastiani, Shynaray Myrzakul, and Ratbay Myrzakulov. Reconstruction of k-essence inﬂation in Horndeski gravity. Eur. Phys. J. Plus, 132(10):433, 2017.

[95] George Koutsoumbas, Konstantinos Ntrekis, Eleftherios Papantonopoulos, and Em- manuel N. Saridakis. Uniﬁcation of Dark Matter - Dark Energy in Generalized Galileon Theories. JCAP, 02:003, 2018.

[96] Chao-Qiang Geng, Md. Wali Hossain, R. Myrzakulov, M. Sami, and Emmanuel N. Saridakis. Quintessential inﬂation with canonical and noncanonical scalar ﬁelds and Planck 2015 results. Phys. Rev. D, 92(2):023522, 2015.

[97] K. Rezazadeh, K. Karami, and S. Hashemi. Tachyon inﬂation with steep potentials. Phys. Rev. D, 95(10):103506, 2017.

[98] Chao-Qiang Geng, Chung-Chi Lee, M. Sami, Emmanuel N. Saridakis, and Alexei A. Starobinsky. Observational constraints on successful model of quintessential Inﬂa- tion. JCAP, 06:011, 2017.

[99] Saﬁa Ahmad, R. Myrzakulov, and M. Sami. Relic gravitational waves from Quintessential Inﬂation. Phys. Rev. D, 96(6):063515, 2017. BIBLIOGRAPHY 157

[100] Abhineet Agarwal, R. Myrzakulov, M. Sami, and Naveen K. Singh. Quintessential inﬂation in a thawing realization. Phys. Lett. B, 770:200–208, 2017.

[101] M.A. Skugoreva, M. Sami, and N. Jaman. Emergence of cosmological scaling behavior in the asymptotic regime. Phys. Rev. D, 100(4):043512, 2019.

[102] Sudipta Das, Manisha Banerjee, and Nandan Roy. Dynamical System Analysis for Steep Potentials. JCAP, 08:024, 2019.

[103] Gustavo B.F. Lima and Rudnei O. Ramos. Unified early and late Universe cosmology through dissipative effects in steep quintessential inflation potential models. Phys. Rev. D, 100(12):123529, 2019.

[104] Smaragda Lola, Andreas Lymperis, and Emmanuel N. Saridakis. Inﬂation with non-canonical scalar ﬁelds revisited. 5 2020.

[105] Alessandro Melchiorri, Laura Mersini, Carolina J. Odman,¨ and Mark Trodden. The state of the dark energy equation of state. Phys. Rev. D, 68:043509, Aug 2003.

[106] Takeshi Chiba, Takahiro Okabe, and Masahide Yamaguchi. Kinetically driven quintessence. Phys. Rev. D, 62:023511, 2000.

[107] Juan M. Aguirregabiria, Luis P. Chimento, and Ruth Lazkoz. Phantom k-essence cosmologies. Phys. Rev. D, 70:023509, 2004.

[108] Edmund J. Copeland, Mohammad R. Garousi, M. Sami, and Shinji Tsujikawa. What is needed of a tachyon if it is to be the dark energy? Phys. Rev. D, 71:043003, 2005.

[109] Grigoris Panotopoulos. Cosmological constraints on a dark energy model with a non-linear scalar ﬁeld. 6 2006.

[110] Michael Malquarti, Edmund J. Copeland, and Andrew R. Liddle. K-essence and the coincidence problem. Phys. Rev. D, 68:023512, 2003.

[111] Ruth Lazkoz. Rigidity of cosmic acceleration in a class of k-essence cosmologies. Int. J. Mod. Phys. D, 14:635–642, 2005. BIBLIOGRAPHY 158

[112] C. Armendariz-Picon, Viatcheslav F. Mukhanov, and Paul J. Steinhardt. A Dy- namical solution to the problem of a small cosmological constant and late time cosmic acceleration. Phys. Rev. Lett., 85:4438–4441, 2000.

[113] C. Armendariz-Picon, Viatcheslav F. Mukhanov, and Paul J. Steinhardt. Essentials of k essence. Phys. Rev. D, 63:103510, 2001.

[114] Takeshi Chiba. Tracking K-essence. Phys. Rev. D, 66:063514, 2002.

[115] Michael Malquarti, Edmund J Copeland, Andrew R Liddle, and Mark Trodden. A New view of k-essence. Phys. Rev. D, 67:123503, 2003.

[116] Shinji Tsujikawa and M. Sami. A Uniﬁed approach to scaling solutions in a general cosmological background. Phys. Lett. B, 603:113–123, 2004.

[117] Federico Piazza and Shinji Tsujikawa. Dilatonic ghost condensate as dark energy. JCAP, 07:004, 2004.

[118] C. Armendariz-Picon and Eugene A. Lim. Haloes of k-essence. JCAP, 08:007, 2005.

[119] Sanil Unnikrishnan. Can cosmological observations uniquely determine the nature of dark energy ? Phys. Rev. D, 78:063007, 2008.

[120] Malcolm Fairbairn and Michel H.G. Tytgat. Inﬂation from a tachyon ﬂuid? Phys. Lett. B, 546:1–7, 2002.

[121] Daniele A. Steer and F. Vernizzi. Tachyon inflation: Tests and comparison with single scalar field inflation. Phys. Rev. D, 70:043527, 2004.

[122] Sergio del Campo, Ramon Herrera, and Adolfo Toloza. Tachyon Field in Interme- diate Inﬂation. Phys. Rev. D, 79:083507, 2009.

[123] Grigoris Panotopoulos. Detectable primordial non-gaussianities and gravitational waves in k-inﬂation. Phys. Rev. D, 76:127302, 2007.

[124] Luis P. Chimento. Extended tachyon ﬁeld, Chaplygin gas and solvable k-essence cosmologies. Phys. Rev. D, 69:123517, 2004. BIBLIOGRAPHY 159

[125] Nilok Bose and A. S. Majumdar. Uniﬁed model ofk-inﬂation, dark matter, and dark energy. Physical Review D, 80(10), Nov 2009.

[126] Daniele Bertacca, Sabino Matarrese, and Massimo Pietroni. Uniﬁed Dark Matter in Scalar Field Cosmologies. Mod. Phys. Lett. A, 22:2893–2907, 2007.

[127] Christian Armendariz-Picon and Eugene A. Lim. Scale invariance without inﬂation? JCAP, 12:002, 2003.

[128] Daniele Bertacca, Nicola Bartolo, and Sabino Matarrese. Halos of Uniﬁed Dark Matter Scalar Field. JCAP, 05:005, 2008.

[129] Robert J. Scherrer. Purely kinetic k-essence as uniﬁed dark matter. Phys. Rev. Lett., 93:011301, 2004.

[130] Josue De-Santiago, Jorge L. Cervantes-Cota, and David Wands. Cosmological phase space analysis of the F(X) - V(φ) scalar ﬁeld and bouncing solutions. Phys. Rev. D, 87(2):023502, 2013.

[131] S.L. Glashow. Partial Symmetries of Weak Interactions. Nucl. Phys., 22:579–588, 1961.

[132] Steven Weinberg. A Model of Leptons. Phys. Rev. Lett., 19:1264–1266, 1967.

[133] Abdus Salam. Weak and Electromagnetic Interactions. Conf. Proc. C, 680519:367– 377, 1968.

[134] Peter W. Higgs. Broken symmetries, massless particles and gauge ﬁelds. Phys. Lett., 12:132–133, 1964.

[135] Peter W. Higgs. Broken Symmetries and the Masses of Gauge Bosons. Phys. Rev. Lett., 13:508–509, 1964.

[136] F. Englert and R. Brout. Broken Symmetry and the Mass of Gauge Vector Mesons. Phys. Rev. Lett., 13:321–323, 1964.

[137] Georges Aad et al. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B, 716:1–29, 2012. BIBLIOGRAPHY 160

[138] Serguei Chatrchyan et al. Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC. Phys. Lett. B, 716:30–61, 2012.

[139] J. Wess and B. Zumino. Supergauge Transformations in Four-Dimensions. Nucl. Phys. B, 70:39–50, 1974.

[140] Th. Kaluza. Zum Unitatsproblem der Physik. Int. J. Mod. Phys. D, 27(14):1870001, 2018.

[141] F. Zwicky. Die Rotverschiebung von extragalaktischen Nebeln. Helv. Phys. Acta, 6:110–127, 1933.

[142] Edvige Corbelli and Paolo Salucci. The Extended Rotation Curve and the Dark Matter Halo of M33. Mon. Not. Roy. Astron. Soc., 311:441–447, 2000.

[143] Richard Massey, Thomas Kitching, and Johan Richard. The dark matter of gravitational lensing. Rept. Prog. Phys., 73:086901, 2010.

[144] Douglas Clowe, Anthony Gonzalez, and Maxim Markevitch. Weak lensing mass reconstruction of the interacting cluster 1E0657-558: Direct evidence for the existence of dark matter. Astrophys. J., 604:596–603, 2004.

[145] Douglas Clowe, Marusa Bradac, Anthony H. Gonzalez, Maxim Markevitch, Scott W. Randall, Christine Jones, and Dennis Zaritsky. A direct empirical proof of the existence of dark matter. Astrophys. J. Lett., 648:L109–L113, 2006.

[146] Andrew J. Benson, Arya Farahi, Shaun Cole, Leonidas A. Moustakas, Adrian Jenk- ins, Mark Lovell, Rachel Kennedy, John Helly, and Carlos Frenk. Dark Matter Halo Merger Histories Beyond Cold Dark Matter: I - Methods and Application to Warm Dark Matter. Mon. Not. Roy. Astron. Soc., 428:1774, 2013.

[147] Mark R. Lovell, Carlos S. Frenk, Vincent R. Eke, Adrian Jenkins, Liang Gao, and Tom Theuns. The properties of warm dark matter haloes. Mon. Not. Roy. Astron. Soc., 439:300–317, 2014.

[148] Rachel Kennedy, Carlos Frenk, Shaun Cole, and Andrew Benson. Constraining the warm dark matter particle mass with Milky Way satellites. Mon. Not. Roy. Astron. Soc., 442(3):2487–2495, 2014. BIBLIOGRAPHY 161

[149] Kim Griest. Galactic Microlensing as a Method of Detecting Massive Compact Halo Objects. Astrophys. J., 366:412–421, 1991.

[150] C. Alcock et al. The MACHO project: Microlensing results from 5.7 years of LMC observations. Astrophys. J., 542:281–307, 2000.

[151] P. Tisserand et al. Limits on the Macho Content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds. Astron. Astrophys., 469:387–404, 2007.

[152] S. Calchi Novati, L. Mancini, G. Scarpetta, and L. Wyrzykowski. LMC self lensing for OGLE-II microlensing observations. Mon. Not. Roy. Astron. Soc., 400:1625, 2009.

[153] L. Wyrzykowski et al. The OGLE View of Microlensing towards the Magellanic Clouds. I. A Trickle of Events in the OGLE-II LMC data. Mon. Not. Roy. Astron. Soc., 397:1228–1242, 2009.

[154] Sebastiano Calchi Novati et al. POINT-AGAPE pixel lensing survey of M31: Ev- idence for a MACHO contribution to galactic halos. Astron. Astrophys., 443:911, 2005.

[155] S.W. Hawking. Black hole explosions. Nature, 248:30–31, 1974.

[156] P. Sreekumar et al. EGRET observations of the extragalactic gamma-ray emission. Astrophys. J., 494:523–534, 1998.

[157] B.J. Carr, Kazunori Kohri, Yuuiti Sendouda, and Jun’ichi Yokoyama. New cosmological constraints on primordial black holes. Phys. Rev. D, 81:104019, 2010.

[158] Fabio Capela, Maxim Pshirkov, and Peter Tinyakov. Constraints on Primordial Black Holes as Dark Matter Candidates from Star Formation. Phys. Rev. D, 87(2):023507, 2013.

[159] A. Barnacka, J.-F. Glicenstein, and R. Moderski. New constraints on primordial black holes abundance from femtolensing of gamma-ray bursts. Physical Review D, 86(4), Aug 2012. BIBLIOGRAPHY 162

[160] Kim Griest, Agnieszka M. Cieplak, and Matthew J. Lehner. Experimental Limits on Primordial Black Hole Dark Matter from the First 2 yr of Kepler Data. Astrophys. J., 786(2):158, 2014.

[161] Massimo Ricotti, Jeremiah P. Ostriker, and Katherine J. Mack. Eﬀect of Primordial Black Holes on the Cosmic Microwave Background and Cosmological Parameter Estimates. Astrophys. J., 680:829, 2008.

[162] Miguel A. Monroy-Rodrguez and Christine Allen. The end of the macho era, revisited: New limits on macho masses from halo wide binaries. The Astrophysical Journal, 790(2):159, Jul 2014.

[163] Bernard J. Carr and S.W. Hawking. Black holes in the early Universe. Mon. Not. Roy. Astron. Soc., 168:399–415, 1974.

[164] P. Meszaros. The behaviour of point masses in an expanding cosmological substra- tum. Astron. Astrophys., 37:225–228, 1974.

[165] Don N. Page and S.W. Hawking. Gamma rays from primordial black holes. Astro- phys. J., 206:1–7, 1976.

[166] B.P. Abbott et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett., 116(6):061102, 2016.

[167] Anne M. Green. Microlensing and dynamical constraints on primordial black hole dark matter with an extended mass function. Phys. Rev. D, 94(6):063530, 2016.

[168] Florian Khnel and Katherine Freese. Constraints on Primordial Black Holes with Extended Mass Functions. Phys. Rev. D, 95(8):083508, 2017.

[169] Bernard Carr, Martti Raidal, Tommi Tenkanen, Ville Vaskonen, and Hardi Veer- mae. Primordial black hole constraints for extended mass functions. Phys. Rev. D, 96(2):023514, 2017.

[170] Nicola Bellomo, Jos Luis Bernal, Alvise Raccanelli, and Licia Verde. Primordial Black Holes as Dark Matter: Converting Constraints from Monochromatic to Ex- tended Mass Distributions. JCAP, 01:004, 2018. BIBLIOGRAPHY 163

[171] Guillermo Ballesteros and Marco Taoso. Primordial black hole dark matter from single ﬁeld inﬂation. Phys. Rev. D, 97(2):023501, 2018.

[172] Sebastien Clesse and Juan Garca-Bellido. Seven Hints for Primordial Black Hole Dark Matter. Phys. Dark Univ., 22:137–146, 2018.

[173] Gian F. Giudice, Matthew McCullough, and Alfredo Urbano. Hunting for Dark Particles with Gravitational Waves. JCAP, 10:001, 2016.

[174] Gianfranco Bertone, Dan Hooper, and Joseph Silk. Particle dark matter: Evidence, candidates and constraints. Phys. Rept., 405:279–390, 2005.

[175] Scott Dodelson and Lawrence M. Widrow. Sterile-neutrinos as dark matter. Phys. Rev. Lett., 72:17–20, 1994.

[176] H.J. de Vega and N.G. Sanchez. Model independent analysis of dark matter points to a particle mass at the keV scale. Mon. Not. Roy. Astron. Soc., 404:885, 2010.

[177] H.J. de Vega, P. Salucci, and N.G. Sanchez. The mass of the dark matter particle from theory and observations. New Astron., 17:653–666, 2012.

[178] Alexander Merle. keV Neutrino Model Building. Int. J. Mod. Phys. D, 22:1330020, 2013.

[179] R. D. Peccei and Helen R. Quinn. CP conservation in the presence of pseudopar- ticles. Phys. Rev. Lett., 38:1440–1443, Jun 1977.

[180] David J. E. Marsh. Axion Cosmology. Phys. Rept., 643:1–79, 2016.

[181] John Preskill, Mark B. Wise, and Frank Wilczek. Cosmology of the invisible axion. Physics Letters B, 120(1):127 – 132, 1983.

[182] L.F. Abbott and P. Sikivie. A cosmological bound on the invisible axion. Physics Letters B, 120(1):133 – 136, 1983.

[183] Michael Dine and Willy Fischler. The not-so-harmless axion. Physics Letters B, 120(1):137 – 141, 1983. BIBLIOGRAPHY 164

[184] C. Abel et al. Search for Axionlike Dark Matter through Nuclear Spin Precession in Electric and Magnetic Fields. Phys. Rev. X, 7(4):041034, 2017.

[185] Edward W. Kolb and Michael S. Turner. The Early Universe, volume 69. 1990.

[186] Kim Griest and Marc Kamionkowski. Unitarity limits on the mass and radius of dark-matter particles. Phys. Rev. Lett., 64:615–618, Feb 1990.

[187] Kim Griest and Marc Kamionkowski. Unitarity Limits on the Mass and Radius of Dark Matter Particles. Phys. Rev. Lett., 64:615, 1990.

[188] Alexander Kusenko, Vadim Kuzmin, Mikhail E. Shaposhnikov, and P.G. Tinyakov. Experimental signatures of supersymmetric dark matter Q balls. Phys. Rev. Lett., 80:3185–3188, 1998.

[189] Alexander Kusenko and Mikhail E. Shaposhnikov. Supersymmetric Q balls as dark matter. Phys. Lett. B, 418:46–54, 1998.

[190] Robert Foot. Implications of the DAMA and CRESST experiments for mirror matter type dark matter. Phys. Rev. D, 69:036001, 2004.

[191] Robert Foot and Z.K. Silagadze. Supernova explosions, 511-keV photons, gamma ray bursts and mirror matter. Int. J. Mod. Phys. D, 14:143–152, 2005.

[192] H.M. Hodges. Mirror baryons as the dark matter. Phys. Rev. D, 47:456–459, 1993.

[193] A.Yu. Ignatiev and R.R. Volkas. Mirror dark matter and large scale structure. Phys. Rev. D, 68:023518, 2003.

[194] R.N. Mohapatra, S. Nussinov, and V.L. Teplitz. Mirror matter as selﬁnteracting dark matter. Phys. Rev. D, 66:063002, 2002.

[195] A. De Rujula, S.L. Glashow, and U. Sarid. CHARGED DARK MATTER. Nucl. Phys. B, 333:173–194, 1990.

[196] Gary Shiu and Lian-Tao Wang. D matter. Phys. Rev. D, 69:126007, 2004.

[197] Edmund J. Copeland, M. Sami, and Shinji Tsujikawa. Dynamics of dark energy. Int. J. Mod. Phys., D15:1753–1936, 2006. BIBLIOGRAPHY 165

[198] Varun Sahni and Alexei A. Starobinsky. The Case for a positive cosmological Lambda term. Int. J. Mod. Phys. D, 9:373–444, 2000.

[199] Varun Sahni. Dark matter and dark energy. Lect. Notes Phys., 653:141–180, 2004.

[200] Harvey B. Richer et al. The Lower main sequence and mass function of the globular cluster Messier 4. Astrophys. J. Lett., 574:L151–L154, 2002.

[201] Brad M.S. Hansen, James Brewer, Greg G. Fahlman, Brad K. Gibson, Rodrigo Ibata, Marco Limongi, R.Michael Rich, Harvey B. Richer, Michael M. Shara, and Peter B. Stetson. The white dwarf cooling sequence of the globular cluster messier 4. Astrophys. J. Lett., 574:L155–L158, 2002.

[202] D.N. Spergel et al. Wilkinson Microwave Anisotropy Probe (WMAP) three year results: implications for cosmology. Astrophys. J. Suppl., 170:377, 2007.

[203] Max Tegmark et al. Cosmological parameters from SDSS and WMAP. Phys. Rev. D, 69:103501, 2004.

[204] Uros Seljak et al. Cosmological parameter analysis including SDSS Ly-alpha forest and galaxy bias: Constraints on the primordial spectrum of ﬂuctuations, neutrino mass, and dark energy. Phys. Rev. D, 71:103515, 2005.

[205] G. Aldering et al. Overview of the Supernova/Acceleration Probe (SNAP). Proc. SPIE Int. Soc. Opt. Eng., 4835:146, 2002.

[206] Steven Weinberg. The Cosmological Constant Problem. Rev. Mod. Phys., 61:1–23, 1989.

[207] G.W. Gibbons, S.W. Hawking, and S.T.C. Siklos, editors. THE VERY EARLY UNIVERSE. PROCEEDINGS, NUFFIELD WORKSHOP, CAMBRIDGE, UK, JUNE 21 - JULY 9, 1982, 1 1984.

[208] J.David Brown and C. Teitelboim. Neutralization of the Cosmological Constant by Membrane Creation. Nucl. Phys. B, 297:787–836, 1988.

[209] John D. Barrow and Frank J. Tipler. The Anthropic Cosmological Principle. Oxford U. Pr., Oxford, 1988. BIBLIOGRAPHY 166

[210] J. Garriga and A. Vilenkin. Solutions to the cosmological constant problems. Phys. Rev. D, 64:023517, 2001.

[211] Jaume Garriga, Andrei Linde, and Alexander Vilenkin. Dark energy equation of state and anthropic selection. Phys. Rev. D, 69:063521, Mar 2004.

[212] E. Baum. Zero Cosmological Constant from Minimum Action. Phys. Lett. B, 133:185–186, 1983.

[213] S.W. Hawking. The Cosmological Constant Is Probably Zero. Phys. Lett. B, 134:403, 1984.

[214] Sidney R. Coleman. Why There Is Nothing Rather Than Something: A Theory of the Cosmological Constant. Nucl. Phys. B, 310:643–668, 1988.

[215] Shamit Kachru, Michael Schulz, and Eva Silverstein. Self-tuning ﬂat domain walls in 5d gravity and string theory. Phys. Rev. D, 62:045021, Jul 2000.

[216] Nima Arkani-Hamed, Savas Dimopoulos, Nemanja Kaloper, and Raman Sundrum. A Small cosmological constant from a large extra dimension. Phys. Lett. B, 480:193– 199, 2000.

[217] Pietro Fre, Mario Trigiante, and Antoine Van Proeyen. Stable de Sitter vacua from N=2 supergravity. Class. Quant. Grav., 19:4167–4194, 2002.

[218] Y. Aghababaie, C.P. Burgess, S.L. Parameswaran, and F. Quevedo. Towards a naturally small cosmological constant from branes in 6-D supergravity. Nucl. Phys. B, 680:389–414, 2004.

[219] Raphael Bousso and Joseph Polchinski. Quantization of four form ﬂuxes and dynamical neutralization of the cosmological constant. JHEP, 06:006, 2000.

[220] Eva Silverstein. (A)dS backgrounds from asymmetric orientifolds. Clay Mat. Proc., 1:179, 2002.

[221] Alexander Maloney, Eva Silverstein, and Andrew Strominger. De Sitter space in noncritical string theory. In Workshop on Conference on the Future of Theoretical BIBLIOGRAPHY 167

Physics and Cosmology in Honor of Steven Hawking’s 60th Birthday, pages 570– 591, 5 2002.

[222] Jonathan L. Feng, John March-Russell, Savdeep Sethi, and Frank Wilczek. Salta- tory relaxation of the cosmological constant. Nucl. Phys. B, 602:307–328, 2001.

[223] C.P. Burgess, R. Kallosh, and F. Quevedo. De Sitter string vacua from supersymmetric D terms. JHEP, 10:056, 2003.

[224] T. Padmanabhan. Vacuum ﬂuctuations of energy density can lead to the observed cosmological constant. Class. Quant. Grav., 22:L107–L110, 2005.

[225] G W Gibbons. Cosmological evolution of the rolling tachyon. Phys. Lett. B, 537:1–4, 2002.

[226] T. Padmanabhan. Accelerated expansion of the universe driven by tachyonic matter. Phys. Rev. D, 66:021301, 2002.

[227] J.S. Bagla, Harvinder Kaur Jassal, and T. Padmanabhan. Cosmology with tachyon ﬁeld as dark energy. Phys. Rev. D, 67:063504, 2003.

[228] L.Raul W. Abramo and Fabio Finelli. Cosmological dynamics of the tachyon with an inverse power-law potential. Phys. Lett. B, 575:165–171, 2003.

[229] J.M. Aguirregabiria and Ruth Lazkoz. Tracking solutions in tachyon cosmology. Phys. Rev. D, 69:123502, 2004.

[230] R.R. Caldwell. A Phantom menace? Phys. Lett. B, 545:23–29, 2002.

[231] Alexander Yu. Kamenshchik, Ugo Moschella, and Vincent Pasquier. An Alternative to quintessence. Phys. Lett. B, 511:265–268, 2001.

[232] Robert J. Scherrer. Phantom dark energy, cosmic doomsday, and the coincidence problem. Phys. Rev., D71:063519, 2005.

[233] S. Nesseris and Leandros Perivolaropoulos. The Fate of bound systems in phantom and quintessence cosmologies. Phys. Rev., D70:123529, 2004. BIBLIOGRAPHY 168

[234] Leandros Perivolaropoulos. Constraints on linear negative potentials in quintessence and phantom models from recent supernova data. Phys. Rev., D71:063503, 2005.

[235] A. Lykkas and L. Perivolaropoulos. Scalar-Tensor Quintessence with a linear potential: Avoiding the Big Crunch cosmic doomsday. Phys. Rev., D93(4):043513, 2016.

[236] M. Eckstein, M. Heller, S.J. Szybka (eds.). Mathematical Structures of the Universe. Copernicus Center Press, Krakow, Poland, pages 99–117, 2014.

[237] Mariusz P. Dabrowski. Are singularities the limits of cosmology? 2014.

[238] Ignatios Antoniadis, J. Rizos, and K. Tamvakis. Singularity - free cosmological solutions of the superstring eﬀective action. Nucl. Phys., B415:497–514, 1994.

[239] Gary N. Felder, Andrei V. Frolov, Lev Kofman, and Andrei D. Linde. Cosmology with negative potentials. Phys. Rev., D66:023507, 2002.

[240] John D. Barrow. Sudden future singularities. Class. Quant. Grav., 21:L79–L82, 2004.

[241] L. Fernandez-Jambrina and Ruth Lazkoz. Geodesic behaviour of sudden future singularities. Phys. Rev., D70:121503, 2004.

[242] David F. Mota and Douglas J. Shaw. Evading Equivalence Principle Violations, Cosmological and other Experimental Constraints in Scalar Field Theories with a Strong Coupling to Matter. Phys. Rev., D75:063501, 2007.

[243] V. K. Onemli and R. P. Woodard. Quantum eﬀects can render w ¡ -1 on cosmological scales. Phys. Rev., D70:107301, 2004.

[244] Christopher J. Fewster and Gregory J. Galloway. Singularity theorems from weak- ened energy conditions. Class. Quant. Grav., 28:125009, 2011.

[245] Ying-li Zhang and Misao Sasaki. Screening of cosmological constant in non-local cosmology. Int. J. Mod. Phys., D21:1250006, 2012. BIBLIOGRAPHY 169

[246] Kazuharu Bamba, Shinichi Nojiri, Sergei D. Odintsov, and Misao Sasaki. Screening of cosmological constant for De Sitter Universe in non-local gravity, phantom-divide crossing and ﬁnite-time future singularities. Gen. Rel. Grav., 44:1321–1356, 2012.

[247] Shin’ichi Nojiri, Sergei D. Odintsov, Misao Sasaki, and Ying-li Zhang. Screening of cosmological constant in non-local gravity. Phys. Lett., B696:278–282, 2011.

[248] Kazuharu Bamba, Ratbay Myrzakulov, Shin’ichi Nojiri, and Sergei D. Odintsov. Reconstruction of f(T ) gravity: Rip cosmology, ﬁnite-time future singularities and thermodynamics. Phys. Rev., D85:104036, 2012.

[249] John D. Barrow, Antonio B. Batista, Julio C. Fabris, Mahouton J. S. Houndjo, and Giuseppe Dito. Sudden singularities survive massive quantum particle production. Phys. Rev., D84:123518, 2011.

[250] Mariam Bouhmadi-Lopez, Yaser Tavakoli, and Paulo Vargas Moniz. Appeasing the Phantom Menace? JCAP, 1004:016, 2010.

[251] Mariam Bouhmadi-Lopez, Che-Yu Chen, and Pisin Chen. Eddington Born Infeld cosmology: a cosmographic approach, a tale of doomsdays and the fate of bound structures. Eur. Phys. J., C75:90, 2015.

[252] Mariam Bouhmadi-Lopez, Claus Kiefer, Barbara Sandhofer, and Paulo Vargas Mo- niz. On the quantum fate of singularities in a dark-energy dominated universe. Phys. Rev., D79:124035, 2009.

[253] Alexander Kamenshchik and Serena Manti. Classical and Quantum big Brake Cos- mology for Scalar Field and Tachyonic Models. In Proceedings, 13th Marcel Gross- mann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG13): Stockholm, Swe- den, July 1-7, 2012, pages 1646–1648, 2015.

[254] A. Kamenshchik and S. Manti. Classical and quantum Big Brake cosmology for scalar ﬁeld and tachyonic models. AIP Conf. Proc., 1514:179–182, 2013.

[255] Alexander Y. Kamenshchik and Serena Manti. Classical and quantum Big Brake cosmology for scalar ﬁeld and tachyonic models. Phys. Rev., D85:123518, 2012. BIBLIOGRAPHY 170

[256] Mariusz P. Dabrowski, Claus Kiefer, and Barbara Sandhofer. Quantum phantom cosmology. Phys. Rev., D74:044022, 2006.

[257] Mariusz P. Dabrowski, Konrad Marosek, and Adam Balcerzak. Standard and exotic singularities regularized by varying constants. Mem. Soc. Ast. It., 85(1):44–49, 2014.

[258] L. Fernandez-Jambrina and Ruth Lazkoz. Singular fate of the universe in modiﬁed theories of gravity. Phys. Lett., B670:254–258, 2009.

[259] Alexander Kamenshchik, Claus Kiefer, and Barbara Sandhofer. Quantum cosmology with big-brake singularity. Phys. Rev., D76:064032, 2007.

[260] Shin’ichi Nojiri and Sergei D. Odintsov. The Future evolution and ﬁnite-time singularities in F(R)-gravity unifying the inﬂation and cosmic acceleration. Phys. Rev., D78:046006, 2008.

[261] Shin’ichi Nojiri and Sergei D. Odintsov. Is the future universe singular: Dark Matter versus modiﬁed gravity? Phys. Lett., B686:44–48, 2010.

[262] M. Sami, Parampreet Singh, and Shinji Tsujikawa. Avoidance of future singularities in loop quantum cosmology. Phys. Rev., D74:043514, 2006.

[263] Parampreet Singh and Francesca Vidotto. Exotic singularities and spatially curved Loop Quantum Cosmology. Phys. Rev., D83:064027, 2011.

[264] R. Penrose. Singularities and big-bang cosmology. Q. J. Roy. Astron. Soc., 29:61– 63, 1988.

[265] F. Briscese, E. Elizalde, S. Nojiri, and S. D. Odintsov. Phantom scalar dark energy as modiﬁed gravity: Understanding the origin of the Big Rip singularity. Phys. Lett., B646:105–111, 2007.

[266] Luis P. Chimento and Ruth Lazkoz. On big rip singularities. Mod. Phys. Lett., A19:2479–2484, 2004. BIBLIOGRAPHY 171

[267] Paul H. Frampton, Kevin J. Ludwick, Shin’ichi Nojiri, Sergei D. Odintsov, and Robert J. Scherrer. Models for Little Rip Dark Energy. Phys. Lett., B708:204–211, 2012.

[268] Paul H. Frampton, Kevin J. Ludwick, and Robert J. Scherrer. Pseudo-rip: Cosmo- logical models intermediate between the cosmological constant and the little rip. Phys. Rev., D85:083001, 2012.

[269] Imogen P. C. Heard and David Wands. Cosmology with positive and negative exponential potentials. Class. Quant. Grav., 19:5435–5448, 2002.

[270] S. Elitzur, A. Giveon, D. Kutasov, and E. Rabinovici. From big bang to big crunch and beyond. JHEP, 06:017, 2002.

[271] Roberto Giambo, John Miritzis, and Koralia Tzanni. Negative potentials and col- lapsing universes II. Class. Quant. Grav., 32(16):165017, 2015.

[272] Vittorio Gorini, Alexander Yu. Kamenshchik, Ugo Moschella, and Vincent Pasquier. Tachyons, scalar ﬁelds and cosmology. Phys. Rev., D69:123512, 2004.

[273] Tomasz Denkiewicz. Observational Constraints on Finite Scale Factor Singularities. JCAP, 1207:036, 2012.

[274] Mariam Bouhmadi-L´opez, Pedro F Gonz´alez-D´ıaz, and Prado Mart´ın-Moruno. Worse than a big rip? Physics Letters B, 659(1):1–5, 2008.

[275] Mariusz P. Dabrowski and Tomasz Denkieiwcz. Barotropic index w-singularities in cosmology. Phys. Rev., D79:063521, 2009.

[276] L. Fernandez-Jambrina. w-cosmological singularities. Phys. Rev., D82:124004, 2010.

[277] L. Fernandez-Jambrina and R. Lazkoz. Classiﬁcation of cosmological milestones. Phys. Rev., D74:064030, 2006.

[278] Frank J. Tipler. Singularities in conformally ﬂat spacetimes. Phys. Lett., A64:8–10, 1977.

[279] A. Krolak. . Class. Quan. Grav., 3:267, 1986. BIBLIOGRAPHY 172

[280] Leandros Perivolaropoulos. Fate of bound systems through sudden future singularities. Phys. Rev., D94(12):124018, 2016.

[281] Wieslaw Rudnicki, Robert J. Budzynski, and Witold Kondracki. Generalized strong curvature singularities and cosmic censorship. Mod. Phys. Lett., A17:387–397, 2002.

[282] Wieslaw Rudnicki, Robert J. Budzynski, and Witold Kondracki. Generalized strong curvature singularities and weak cosmic censorship in cosmological space-times. Mod. Phys. Lett., A21:1501–1510, 2006.

[283] Shin’ichi Nojiri, Sergei D. Odintsov, and Shinji Tsujikawa. Properties of singularities in the (phantom) dark energy universe. Phys. Rev. D, 71:063004, Mar 2005.

[284] J. D. Barrow, G. J. Galloway, and F. J. Tipler. The closed-universe recollapse conjecture. MNRAS, 223:835–844, December 1986.

[285] John D. Barrow, S. Cotsakis, and A. Tsokaros. A General Sudden Cosmological Singularity. Class. Quant. Grav., 27:165017, 2010.

[286] Mariusz P. Dabrowski and Konrad Marosek. Regularizing cosmological singularities by varying physical constants. JCAP, 1302:012, 2013.

[287] Konrad Marosek, Mariusz P. Dabrowski, and Adam Balcerzak. Cyclic multiverses. Mon. Not. Roy. Astron. Soc., 461(3):2777–2788, 2016.

[288] A. Kamenshchik, Zoltan Keresztes, and Laszlo A. Gergely. The paradox of soft singularity crossing avoided by distributional cosmological quantities. In Proceedings, 13th Marcel Grossmann Meeting on Recent Developments in Theoretical and Exper- imental General Relativity, Astrophysics, and Relativistic Field Theories (MG13): Stockholm, Sweden, July 1-7, 2012, pages 1847–1849, 2015.

[289] Zoltan Keresztes, Laszlo A. Gergely, and Alexander Yu. Kamenshchik. The paradox of soft singularity crossing and its resolution by distributional cosmological quantitities. Phys. Rev., D86:063522, 2012.

[290] Georgios Koﬁnas, Roy Maartens, and Eleftherios Papantonopoulos. Brane cosmology with curvature corrections. Journal of High Energy Physics, 2003(10):066, 2003. BIBLIOGRAPHY 173

[291] Gianluca Calcagni. Slow roll parameters in braneworld cosmologies. Phys. Rev., D69:103508, 2004.

[292] John D. Barrow and Alexander A. H. Graham. Singular Inﬂation. Phys. Rev., D91(8):083513, 2015.

[293] John D. Barrow and Alexander A. H. Graham. New Singularities in Unexpected Places. Int. J. Mod. Phys., D24(12):1544012, 2015.

[294] Salvatore Capozziello. Curvature quintessence. Int. J. Mod. Phys., D11:483–492, 2002.

[295] Antonio De Felice and Shinji Tsujikawa. f(R) theories. Living Rev. Rel., 13:3, 2010.

[296] Shin’ichi Nojiri and Sergei D. Odintsov. Modiﬁed Gauss-Bonnet theory as gravitational alternative for dark energy. Phys. Lett., B631:1–6, 2005.

[297] Antonio De Felice and Shinji Tsujikawa. Construction of cosmologically viable f(G) dark energy models. Phys. Lett., B675:1–8, 2009.

[298] D. Lovelock. The Einstein tensor and its generalizations. J. Math. Phys., 12:498– 501, 1971.

[299] Nathalie Deruelle and Luis Farina-Busto. The Lovelock Gravitational Field Equa- tions in Cosmology. Phys. Rev., D41:3696, 1990.

[300] Philip D. Mannheim and Demosthenes Kazanas. Exact Vacuum Solution to Con- formal Weyl Gravity and Galactic Rotation Curves. Astrophys. J., 342:635–638, 1989.

[301] Eanna E. Flanagan. Fourth order Weyl gravity. Phys. Rev., D74:023002, 2006.

[302] Alberto Nicolis, Riccardo Rattazzi, and Enrico Trincherini. The Galileon as a local modiﬁcation of gravity. Phys. Rev., D79:064036, 2009.

[303] C. Deﬀayet, Gilles Esposito-Farese, and A. Vikman. Covariant Galileon. Phys. Rev., D79:084003, 2009. BIBLIOGRAPHY 174

[304] Genly Leon and Emmanuel N. Saridakis. Dynamical analysis of generalized Galileon cosmology. JCAP, 1303:025, 2013.

[305] Gabriel R. Bengochea and Rafael Ferraro. Dark torsion as the cosmic speed-up. Phys. Rev. D, 79:124019, 2009.

[306] Eric V. Linder. Einstein’s Other Gravity and the Acceleration of the Universe. Phys. Rev., D81:127301, 2010. [Erratum: Phys. Rev.D82,109902(2010)].

[307] Shih-Hung Chen, James B. Dent, Sourish Dutta, and Emmanuel N. Saridakis. Cosmological perturbations in f(T) gravity. Phys. Rev., D83:023508, 2011.

[308] Georgios Koﬁnas and Emmanuel N. Saridakis. Teleparallel equivalent of Gauss- Bonnet gravity and its modiﬁcations. Phys. Rev., D90:084044, 2014.

[309] Georgios Koﬁnas and Emmanuel N. Saridakis. Cosmological applications of

F (T,TG) gravity. Phys. Rev., D90:084045, 2014.

[310] Ted Jacobson. Thermodynamics of space-time: The Einstein equation of state. Phys. Rev. Lett., 75:1260–1263, 1995.

[311] T. Padmanabhan. Gravity and the thermodynamics of horizons. Phys. Rept., 406:49–125, 2005.

[312] T. Padmanabhan. Thermodynamical Aspects of Gravity: New insights. Rept. Prog. Phys., 73:046901, 2010.

[313] Andrei V. Frolov and Lev Kofman. Inﬂation and de Sitter thermodynamics. JCAP, 0305:009, 2003.

[314] Rong-Gen Cai and Sang Pyo Kim. First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe. JHEP, 02:050, 2005.

[315] M. Akbar and Rong-Gen Cai. Thermodynamic Behavior of Friedmann Equations at Apparent Horizon of FRW Universe. Phys. Rev., D75:084003, 2007.

[316] Rong-Gen Cai and Li-Ming Cao. Uniﬁed ﬁrst law and thermodynamics of apparent horizon in FRW universe. Phys. Rev., D75:064008, 2007. BIBLIOGRAPHY 175

[317] M. Akbar and Rong-Gen Cai. Friedmann equations of FRW universe in scalar- tensor gravity, f(R) gravity and ﬁrst law of thermodynamics. Phys. Lett., B635:7– 10, 2006.

[318] Aseem Paranjape, Sudipta Sarkar, and T. Padmanabhan. Thermodynamic route to ﬁeld equations in Lancos-Lovelock gravity. Phys. Rev., D74:104015, 2006.

[319] Ahmad Sheykhi, Bin Wang, and Rong-Gen Cai. Thermodynamical properties of apparent horizon in warped DGP braneworld. Nucl. Phys., B779:1–12, 2007.

[320] Mubasher Jamil, Emmanuel N. Saridakis, and M. R. Setare. Thermodynamics of dark energy interacting with dark matter and radiation. Phys. Rev., D81:023007, 2010.

[321] Rong-Gen Cai and Nobuyoshi Ohta. Horizon Thermodynamics and Gravitational Field Equations in Horava-Lifshitz Gravity. Phys. Rev., D81:084061, 2010.

[322] Mengjie Wang, Jiliang Jing, Chikun Ding, and Songbai Chen. First law of thermodynamics in IR modiﬁed Horava-Lifshitz gravity. Phys. Rev., D81:083006, 2010.

[323] Mubasher Jamil, Emmanuel N. Saridakis, and M. R. Setare. The generalized second law of thermodynamics in Horava-Lifshitz cosmology. JCAP, 1011:032, 2010.

[324] Yongwan Gim, Wontae Kim, and Sang-Heon Yi. The ﬁrst law of thermodynamics in Lifshitz black holes revisited. JHEP, 07:002, 2014.

[325] Zhong-Ying Fan and H. Lu. Thermodynamical First Laws of Black Holes in Quadratically-Extended Gravities. Phys. Rev., D91(6):064009, 2015.

[326] Rong-Gen Cai, Li-Ming Cao, and Ya-Peng Hu. Hawking Radiation of Apparent Horizon in a FRW Universe. Class. Quant. Grav., 26:155018, 2009.

[327] Dongsu Bak and Soo-Jong Rey. Cosmic holography. Class. Quant. Grav., 17:L83, 2000.

[328] Sean A. Hayward. Uniﬁed ﬁrst law of black hole dynamics and relativistic thermodynamics. Class. Quant. Grav., 15:3147–3162, 1998. BIBLIOGRAPHY 176

[329] Sean A. Hayward, Shinji Mukohyama, and M. C. Ashworth. Dynamic black hole entropy. Phys. Lett., A256:347–350, 1999.

[330] G. W. Gibbons and S. W. Hawking. Cosmological Event Horizons, Thermodynam- ics, and Particle Creation. Phys. Rev., D15:2738–2751, 1977.

[331] German Izquierdo and Diego Pavon. Dark energy and the generalized second law. Phys. Lett., B633:420–426, 2006.

[332] C. Brans and R.H. Dicke. Mach’s principle and a relativistic theory of gravitation. Phys. Rev., 124:925–935, 1961.

[333] Luca Amendola. Scaling solutions in general nonminimal coupling theories. Phys. Rev. D, 60:043501, 1999.

[334] Jean-Philippe Uzan. Cosmological scaling solutions of nonminimally coupled scalar ﬁelds. Phys. Rev. D, 59:123510, 1999.

[335] Takeshi Chiba. Quintessence, the gravitational constant, and gravity. Phys. Rev. D, 60:083508, 1999.

[336] Nicola Bartolo and Massimo Pietroni. Scalar tensor gravity and quintessence. Phys. Rev. D, 61:023518, 2000.

[337] Francesca Perrotta, Carlo Baccigalupi, and Sabino Matarrese. Extended quintessence. Phys. Rev. D, 61:023507, 1999.

[338] A. Lymperis, L. Perivolaropoulos, and S. Lola. Sudden Future Singularities in Quintessence and Scalar-Tensor Quintessence Models. Phys. Rev. D, 96(8):084024, 2017.

[339] Andreas Lymperis. Sudden Future Singularities and their observational signatures in Modiﬁed Gravity. PoS, CORFU2017:088, 2018.

[340] L. Perivolaropoulos and C. Sourdis. Cosmological eﬀects of radion oscillations. Phys. Rev., D66:084018, 2002.

[341] L. Perivolaropoulos. Equation of state of oscillating Brans-Dicke scalar and extra dimensions. Phys. Rev., D67:123516, 2003. BIBLIOGRAPHY 177

[342] Matthew C. Johnson and Marc Kamionkowski. Dynamical and Gravitational Insta- bility of Oscillating-Field Dark Energy and Dark Matter. Phys. Rev., D78:063010, 2008.

[343] N. A. Lima, P. T. P. Viana, and I. Tereno. Constraining Recent Oscillations in Quintessence Models with Euclid. Mon. Not. Roy. Astron. Soc., 441(4):3231–3237, 2014.

[344] Sourish Dutta and Robert J. Scherrer. Evolution of Oscillating Scalar Fields as Dark Energy. Phys. Rev., D78:083512, 2008.

[345] John D. Barrow. More general sudden singularities. Class. Quant. Grav., 21:5619– 5622, 2004.

[346] J D Barrow and A C Ottewill. The stability of general relativistic cosmological theory. Journal of Physics A: Mathematical and General, 16(12):2757, 1983.

[347] R. R. Caldwell and Eric V. Linder. The Limits of quintessence. Phys. Rev. Lett., 95:141301, 2005.

[348] Robert J. Scherrer and A. A. Sen. Thawing quintessence with a nearly ﬂat potential. Phys. Rev., D77:083515, 2008.

[349] S. Capozziello, S. Nesseris, and L. Perivolaropoulos. Reconstruction of the Scalar- Tensor Lagrangian from a LCDM Background and Noether Symmetry. JCAP, 0712:009, 2007.

[350] Mariusz P. Dabrowski, Tomasz Denkiewicz, and Martin A. Hendry. How far is it to a sudden future singularity of pressure? Phys. Rev., D75:123524, 2007.

[351] Leandros Perivolaropoulos. Crossing the phantom divide barrier with scalar tensor theories. JCAP, 0510:001, 2005.

[352] S. Nesseris and Leandros Perivolaropoulos. The Limits of Extended Quintessence. Phys. Rev., D75:023517, 2007.

[353] S. Nesseris and Leandros Perivolaropoulos. Crossing the Phantom Divide: Theo- retical Implications and Observational Status. JCAP, 0701:018, 2007. BIBLIOGRAPHY 178

[354] Andreas Lymperis and Emmanuel N. Saridakis. Modiﬁed cosmology through nonextensive horizon thermodynamics. Eur. Phys. J. C, 78(12):993, 2018.

[355] Constantino Tsallis. Possible generalization of boltzmann-gibbs statistics. Journal of Statistical Physics, 52(1):479–487, Jul 1988.

[356] M. L. Lyra and C. Tsallis. Nonextensivity and multifractality in low-dimensional dissipative systems. Phys. Rev. Lett., 80:53–56, Jan 1998.

[357] G. Wilk and Z. Wlodarczyk. On the interpretation of nonextensive parameter q in Tsallis statistics and Levy distributions. Phys. Rev. Lett., 84:2770, 2000.

[358] Ed´esioM. Barboza, Jr., Rafael da C. Nunes, Everton M. C. Abreu, and Jorge Ananias Neto. Dark energy models through nonextensive Tsallis’ statistics. Phys. A, 436:301–310, 2015.

[359] Emmanuel N. Saridakis, Kazuharu Bamba, R. Myrzakulov, and Fotios K. Anagnos- topoulos. Holographic dark energy through Tsallis entropy. JCAP, 1812(12):012, 2018.

[360] Constantino Tsallis and Leonardo J. L. Cirto. Black hole thermodynamical entropy. The European Physical Journal C, 73(7):2487, Jul 2013.

[361] P. A. R. Ade et al. Planck 2015 results. XIII. Cosmological parameters. Astron. Astrophys., 594:A13, 2016.

[362] N. Suzuki et al. The Astrophysical Journal, 746(1):85, jan 2012.

[363] Sheng Li and Andrew R. Liddle. Observational constraints on K-inﬂation models. JCAP, 10:011, 2012.

[364] David I. Kaiser. Primordial spectral indices from generalized Einstein theories. Phys. Rev. D, 52:4295–4306, 1995.

[365] Misao Sasaki and Ewan D. Stewart. A General analytic formula for the spectral index of the density perturbations produced during inﬂation. Prog. Theor. Phys., 95:71–78, 1996. BIBLIOGRAPHY 179

[366] R.P. Woodard. Perturbative Quantum Gravity Comes of Age. Int. J. Mod. Phys. D, 23(09):1430020, 2014.

[367] G. German, A. de la Macorra, and M. Mondragon. Low scale supergravity inﬂation with R symmetry. Phys. Lett. B, 494:311–317, 2000.

[368] Shin’ichi Nojiri and Sergei D. Odintsov. Introduction to modiﬁed gravity and gravitational alternative for dark energy. eConf, C0602061:06, 2006. [Int. J. Geom. Meth. Mod. Phys.4,115(2007)].