Cosmology in Modified Theories of Gravity

Home , Quintom scenario, Sergei Odintsov, Superfluid vacuum theory, Teleparallelism, Varun Sahni

nikolaos th. chatzarakis COSMOLOGYINMODIFIEDTHEORIESOFGRAVITY

Master of Science “Computational Physics” Department of Physics Faculty of Sciences Aristoteleion University of Thessaloniki

COSMOLOGYINMODIFIEDTHEORIESOFGRAVITY A search for viable cosmological models in f (R) and generalised Gauss-Bonnet theories

nikolaos th. chatzarakis

Master of Science “Computational Physics” Department of Physics Faculty of Sciences Aristoteleion University of Thessaloniki

June 2019 – Nikolaos Th. Chatzarakis: Cosmology in Modiﬁed Theories of Gravity, A search for viable cosmological models in f (R) and generalised Gauss-Bonnet theories, © June 2019 “Cosmologists are often in error, but never in doubt.”

Lev Davidovich Landau

“We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the ﬁrst time. Through the unknown, remembered gate When the last of earth left to discover Is that which was the beginning; At the source of the longest river The voice of the hidden waterfall And the children in the apple-tree Not known, because not looked for But heard, half-heard, in the stillness Between two waves of the sea.”

T.S. Eliot, The Four Quartets

“Space may be the ﬁnal frontier but it’s made in a Hollywood basement And Cobain can you hear the spheres singing songs off Station To Station? And Alderaan’s not far away, it’s Californication.”

Red Hot Chilly Peppers, Californication

ABSTRACT

The purpose of this dissertation is to present the most typical modified theories of gravity and discuss their validity and viability in constructing a cosmological model. Current issues in cosmology, related to the Standard Cosmological model, i. e.the “Big Bang model” and the Λ-CDM FLRW model, and including the inflationary scenario, the post-inflation reheating and the late-time accelerating expansion leading to “dark energy”, are considered obstacles for the traditional theory of gravity, i. e.General Theory of Relativity, since they can be explained only by means of exogenous fields a posteriori introduced. The goal of many modified theories of gravity is to extend General Relativity in such a way that quantum or sting-theory corrections ate taken into account, but also in order for many such cosmological -or astrophysical- problems to be resolved endogenously, without any additional hypotheses. We shall briefly mention the key modified theories of gravity, namely the f (R) theory, the f (R, G) theory and the mimetic Einstein- Gauss-Bonnet theory; then, we will proceed by testing their viability in specific cosmological problems, such as the early-time and late-time dynamics. Index terms— f (R) theory of gravity, f (G) theory of gravity, Loop Quantum Cosmology, Inflation, Dark Energy, Reconstruction techniques, Dynamical systems

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PUBLICATIONS

Part of this dissertation has appeared previously in the following publications:

Shin’ichi Nojiri, S.D. Odintsov, V.K. Oikonomou, N. Chatzarakis, and Tan- moy Paul. “Viable Inﬂationary Models in a Ghost-free Gauss-Bonnet The- ory of Gravity.” In: European Physical Journal C (under revision) (2019). arXiv: 1907.00403 [gr-qc].

V. K. Oikonomou and N. Chatzarakis. “The Phase Space of k-Essence f (R) Gravity Theory.” In: (2019). arXiv: 1905.01904 [gr-qc].

ACKNOWLEDGMENTS

The elaboration of a Master Dissertation is usually a hard work compressed in a schedule of some months, perhaps a year, giving the M.Sc. student little time to settle himself/herself into the deep waters of the actual work done. Consequently, much of it is not due to personal work, but rather to enlightenment and encour- agement provided by the supervisor and the other professors. Personally, I would like to thank Prof. C.G. Tsagas for the initial spark in cosmology over three years ago, Prof. N. Stergioulas for any comments, as well as introducing me to Dr. V.K. Oikonomou, and last but not least, dr. V.K. Oikonomou himself for supervising the work and assisting in all parts that demanded so. Through the latter, I would also like to thank Profs. S. Nojiri and S.D. Odintsov for the collaboration. Furthermore, the time and labour expended during the completion of such a Dissertation is often more than originally calculated and demands extreme moral and material support, as well as understanding. In my case, all demands were more that equally met by my parents, my sister and my friends, being not only a continuous resort, but also unnaturally patient. Through this, though not in its absolute measure, I would like to express my graduate to all these people who stood by me, even in times I proved myself not as worthy.

CONTENTS i introduction1 1 introduction3 1.1 The Standard Model and the Cosmological Eras ...... 10 1.2 Cosmic Inflation and Scalar Fields ...... 18 1.3 Accelerating Expansion and Cosmological Constant ...... 26 1.3.1 Canonical Dark Energy ...... 27 1.3.2 Phantom Dark Energy ...... 29 1.3.3 Quintessence ...... 30 1.4 Past and Future Finite-time Singularities ...... 31 1.5 A Notice on Notions and Notation ...... 33 ii a roadmap to modified theories of gracity 35 2 the f (R) theory of gravity 37 2.1 General properties ...... 37 2.1.1 The metric tensor ...... 37 2.1.2 The Affine Connection: torsion-free Levi-Civita case . . . . 39 2.1.3 Curvature ...... 41 2.2 Field equations and Conservation laws ...... 46 2.3 Scalar-tensor description and Brans-Dicke equivalence ...... 50 2.4 Viable f (R) theories ...... 53 2.5 Viable f (R, φ, X) theories ...... 55 2.5.1 Canonical scalar field description ...... 55 2.5.2 Non-canonical scalar field inflation ...... 58 2.5.3 Inflation with f (R, φ) theories of gravity ...... 59 2.6 A cosmological model ...... 61 3 the gauss-bonnet theories of gravity 65 3.1 General Properties ...... 65 3.2 Field equations and Conservation Laws ...... 67 3.2.1 Minimal coupling ...... 69 3.2.2 Non-minimal coupling ...... 70 3.3 “Mimetic” ghost-free theory ...... 72 3.3.1 The Flat FRW Vacuum without ghosts ...... 76 3.4 A Cosmological Model ...... 78 α β 3.4.1 Minimal coupling: f (R, G) = R + f1R + f2G ...... 79 α β 3.4.2 Non-minimal coupling: f (R, G) = f0R G ...... 80 3.4.3 The finite-time singularities: f (R, G) = R + p(t)G + q(t) .. 81 iii recosnstruction of cosmological models 85 4 the f (R) theory under testing 87

xiii xiv contents

4.1 The reconstruction technique ...... 87 4.1.1 Inflationary dynamics of f (R) gravity: Formalism ...... 88 4.2 The deSitter and the quasi-deSitter expansion ...... 91 4.3 The radiation-dominated and the matter-dominated eras ...... 93 4.4 An exponential early-time expansion ...... 95 4.5 A hyperbolic tangent late-time evolution ...... 100 4.6 Conclusions ...... 102 5 the ghost-free einstein-gauss-bonnet theory 107 5.1 Inflationary Dynamics ...... 107 5.2 The deSitter expansion ...... 110 5.2.1 An exponential coupling function, h(χ) = aebχ ...... 111 5.2.2 A power-law coupling function, h(χ) = aχb ...... 114 5.3 The quasi-deSitter expansion ...... 117 5.3.1 An exponential coupling function, h(χ) = aebχ ...... 118 5.3.2 A power-law coupling function, h(χ) = aχb ...... 121 5.3.3 The coupling-free inflation of a quasi-DeSitter background 123 5.4 Conclusions ...... 124

iv phase space analysis of alternative cosmologies 127 6 the interacting fluids description of dark matter and dark energy 129 6.1 The two-fluids description ...... 130 6.2 The Classical case (1): setting up the model ...... 131 6.3 The Classical case (2): working out the model ...... 133 6.3.1 The generalised dark energy ...... 134 6.3.2 The superfluid dark matter and generalised dark energy . . 135 6.3.3 The superfluid dark matter and generalised Chaplygin dark energy ...... 139 6.4 The Loop Quantum Cosmology case (1): setting up the model . . . 143 6.5 The Loop Quantum Cosmology case (2): working out the model . 146 6.5.1 The superfluid dark matter ...... 146 6.5.2 The superfluid dark matter and generalised Chaplygin dark energy ...... 148 6.6 Conclusions ...... 149 7 the phase space of the k-essence theory 153 7.1 Introduction ...... 153 7.2 Setting up the model ...... 154 7.2.1 Equations of Motion of the k-Essence f (R) Gravity Theory 154 7.2.2 The choice of the dynamical variables and their evolution . 155 7.2.3 Friedmann Constraint and the Effective Equation of State . 159 7.2.4 Integrability of the Differential Equations for x3 and x4 ... 161 7.2.5 The three free parameters ...... 163 7.3 Working out the model (1): main analytical and numerical results . 164 contents xv

7.3.1 Model Iα (1): the case of canonical dark energy (c1 = −1) . 164 7.3.2 Model Iα (2): the case of phantom ﬁelds (c1 = 1) ...... 171 7.3.3 Model Iβ: the case of bounce (c1 = 0) ...... 176 7.4 Working out the model (2): additional characteristics ...... 179 7.4.1 Model Iα: a possible 2 − d attractor ...... 179 7.4.2 Model Iβ: a possible 1 − d attractor ...... 181 7.4.3 Inﬁnities encountered in the models ...... 183 7.5 Conclusions ...... 184 v conclusion 187 8 conclusion 189 bibliography 191 LISTOFFIGURES

Figure 1 The phase space of eqs. (20) and (21) for different values of w, corresponding to different types of ﬂuid...... 16 Figure 2 The solution to the Horizon problem...... 21 Figure 3 A typical potential for the scalar ﬁeld, φ...... 24 Figure 4 The spectral index and the scalar-to-tensor ratio for “negative times” with respect to C0 for N = 50 and C1 = C2 = 1. 96 Figure 5 The spectral index and the scalar-to-tensor ratio for “negative times” with respect to C0 for N = 60 and C1 = C2 = 1. 98 Figure 6 The spectral index for “negative times” with respect to HE in the left plot, and to Λ in the right plot, for N = 50, C0 = 47 and C1 = C2 = 1...... 99 Figure 7 The tensor-to-scalar ratio for “negative times” with respect to HE in the left, and to Λ in the right, for N = 50, C0 = 47 and C1 = C2 = 1...... 99 Figure 8 The tensor-to-scalar ratio against the spectral index, for N = 50 in the left, and N = 60 in the right, C1 = C2 = 1, HE = 20 and varying C0...... 104 Figure 9 The tensor-to-scalar ratio against the spectral index, for N = 50 in the left, and N = 60 in the right, C1 = C2 = 1, Λ = 640 and varying C0...... 105 Figure 10 The spectral index nS and the tensor-to-scalar ratio r, with 12 respect to H0, for N = 50 , a = 1 and µ = 10 in the deSitter background with exponential coupling...... 113 Figure 11 The spectral index nS and the tensor-to-scalar ratio r, with respect to µ, for N = 50 and a = 1 in the deSitter background with exponential coupling...... 113 Figure 12 The spectral index nS and the tensor-to-scalar ratio r, with respect to b, for N = 50 , a = 1 and µ = 1012 in the deSitter background with exponential coupling...... 114 Figure 13 The spectral index nS and the tensor-to-scalar ratio r, with respect to b, for N = 50 , a = 1 and µ = 1012 in the deSitter background with power-law coupling...... 117 Figure 14 The spectral index nS with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1 in the quasi-deSitter background with exponential coupling. . . . 120 Figure 15 The tensor-to-scalar ratio with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1 in the quasi-deSitter background with exponential coupling. . . . 120

xvi List of Figures xvii

Figure 16 The spectral index nS with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1 in the quasi-deSitter background with power-law coupling. . . . 123 Figure 17 The tensor-to-scalar ratio with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1 in the quasi-deSitter background with power-law coupling. . . . 123 Figure 18 A parametric plot of the tensor-to-scalar ratio (vertical axis) over the spectral index (horizontal line) for N = 50, a = b = µ = 1 and varying H0 for specified H1 in the quasi- deSitter background...... 124 Figure 19 A parametric plot of the tensor-to-scalar ratio (vertical axis) over the spectral index (horizontal line) for N = 50, a = b = µ = 1 and varying H1 for specified H0 in the quasi- deSitter background...... 125 Figure 20 Intersections of the phase space, for the case of simple generalised fluid dark energy...... 136 Figure 21 Intersections of the phase space, for the case of superfluid dark matter and generalised dark energy...... 138 Figure 22 Intersections of the phase space, for another case of superfluid dark matter and generalised dark energy...... 139 Figure 23 Intersections of the phase space, for a viable case of superfluid dark matter and generalised Chaplyign dark energy. 141 Figure 24 Intersections of the phase space, for an unviable case of superfluid dark matter and generalised Chaplyign dark energy.142 Figure 25 Intersection of the phase space, for the case of superfluid dark matter...... 148 Figure 26 Intersections of the phase space, for a viable case of generalised Chaplyign dark energy...... 150 Figure 27 Analytical solutions derived for the differential equation (403)...... 161 Figure 28 Analytical solutions derived for the differential equation (407)...... 162 Figure 29 2-d intersections of the phase space along the x1 direction, for c1 = −1, m = 0, fD = 3 and x4 = −1...... 166 Figure 30 Several x1-x2 intersections of the phase space, for c1 = −1, 9 m = − , f = 3 and x = −1...... 167 2 D 4 Figure 31 2-d intersections of the phase space along the x1 direction, 9 for c = −1, m = − , f = 3 and x = −1...... 169 1 2 D 4 Figure 32 2-d intersections of the phase space along the x1 direction, for c1 = −1, m = −8, fD = 3...... 172 Figure 33 2-d intersections of the phase space along the x1 direction, 1 for c = 1, m = 0, f = and x = 2, x = −1...... 174 1 D 2 3 4 xviii List of Figures

Figure 34 2-d intersections of the phase space along the x1 direction, 9 1 1 for c = 1, m = − , f = and x = ...... 176 1 2 D 2 3 2 Figure 35 Several x1-x2 intersections of the phase space, for c1 = 1, 1 m = −8 and f = ...... 177 D 2 Figure 36 Analytical solutions of eq. (399) that correspond to the 1 − d attractor existing for c1 = 0...... 182 Part I

INTRODUCTION

INTRODUCTION 1

As of 1916, when the General Theory of Relativity was formulated by A. Ein- stein, 1922 and 1927, when A. Friedmann and G. Lemaître independently reached to a theory for the expansion of the relativistic space-time, and 1929, when E. Hubble confirmed their results by observing that light form distant galaxies was redshifted and thus distant galaxies tended to move away from the Milky Way, indicating that the universe is expanding, the cosmological implications of the relativistic theory of gravity are among the top physical problems issued in this area of research. In fact, questions about the origin and evolution of the space- time, the formation of structure (such as galaxies and galactic clusters), the very early times of the universe (linking to the Big Bang theory), the late-time and future evolution of the universe (linking to the accelerating expansion) and the matter and energy contain of it (especially the nature of dark matter, responsible for structure formation, and of dark energy, responsible for the late-time acceleration) are given extreme attention, since the answer to them might reveal not only the exact “history” of our world, but also the foundations of our physical reality. Questions such as this are inextricably linked to the quest for a unified theory of the of interactions, where gravity, electromagnetism and (weak and strong) nuclear interactions would be combined into one self-coherent force of nature. The development of such a theory is considered essential in order to re-establish New- ton’s idea of a unique theory describing both macro- and micro-scales; a theory that would not distinguish the physics of the large scale, such as in astrophysical and cosmological phenomena, and those of the microscopic processes, such as the nucleon-nucleon interactions or the quark-gluon combinations that con- structs matter. Current research tend to set those aside, with Einstein’s General Relativity describing the gravitational interaction as a consequence of space-time curvature and thus analyzing the large scale astrophysical and cosmological phenomena, and Quantum Field Theory describing the fundamental interactions between matter particles (quarks and leptons) and interaction carriers (photons, Z and W bosons and gluons), whose outcomes correspond to the electromagnetic, weak nuclear and strong nuclear interactions relevant only in the microscopic scales. The specific physical conceptualisation and mathematical formulation of the two, has not so far allowed for a combination under common framework, that on the one hand would realise the quantum fields interactions as geometric notions, and on the other would give gravitational interaction a quantum field formulation. Despite the fact that many frameworks have been proposed from the decades of 1960’s and 1970’s, including the Kaluza-Klein theory, the Brans-Dicke theory,

3 4 introduction

the string and superstring theory, the M-theory, the supersymmetry theory, the AdS/CFT correspondence and the holographic principle, the topological quantum ﬁeld theories, the CGHS and RST models, the Bruch-Davies vacuum theory, the superﬂuid vacuum theory, the causal dynamical triangulation and the Barrett- Crane model and the Loop Quantum Gravity inter alia; none of these has reached a totally accepted and favourable outcome with the majority being non-capable for empirical or experimental tests, due to the high energy levels they require. However, a number of cosmological problems does not fully resolves within the framework of the General Relativity and the absence of a Quantum Theory of Gravity implies that they cannot be resolved (here and now) by means of a fundamental theory. Such problems are usually the following: 1. The existence and the cause of the Big Bang as an initiating point for the space-time as we know it and analyse it by means of the General Relativity; the Big Bang is generally consistent with Friedmann-Lemaître description of the expanding universe as Lemaître’s initial point of the space-time expansion and thus it perceived as the initial conical time-like singularity in General Relativity, which however indicates the limitations of the latter.

2. The subsequent rapid expansion of the space-time, considered to initiate −44 sometime after Planck time (tPl = 5.39116 10 sec) and finish at about 10−32 sec after the initial singularity (the Big Bang); this rapid expansion called cosmic inflation is generally formulated as a deSitter expansion, however it cannot be derived naturally from General Relativity, unless specific scalar fields are assumed.

3. The structure formation and the physics of galaxies and galactic clusters imply the existence of an unobserved non-luminous type of matter, the dark matter, which comprises about 27 % of the total content of the universe; this type of matter, although it is necessary within the current theory of structure formation, it cannot be explained within the framework of the Standard Model for Particle Physics and is introduced in relativistic cosmology and the Friedmann-Lemaître model as an unknown effective ﬁeld.

4. The same holds for dark energy, the remaining 68 % of the total matter- energy content of the universe, which stands for the late-time accelerating expansion of the space-time, observed recently by means of the “standard candles”; many candidates have been proposed for the explanation of this behaviour, such as the emergence of a quantum ﬁeld, the energy of quantum vacuum, or a missing geometric characteristic of the theory of gravity, however a fundamental way to resolve the issue has not yet been found, and in General Relativity the effects of the late-time accelerating expansion is introduced either as an effective ﬁeld, or as the “Cosmological constant”. The above issues are fundamental for the understanding of the universe, however cannot be resolved by fundamental means within the Standard Cosmological introduction 5

Model, as well as the Standard Model for Particle Physics. In fact, it is commonly believed that, as far as these “stylized facts” are true, the unified theory (e.g. a quantum theory of gravity) is essential in further understanding and explaining them. As long as the latter is not found, key notes of resolving these have been found in the various modified gravity theories developed by extending particular aspects of Einstein’s General Theory of Relativity and (partially) linking them with one (or more) Quantum Field Theories, such as the string theory, the M- theory, the Loop Quantum Gravity, etc. A modified theory of gravity is derived by an action, similarly to General Rela- tivity that is derived from the Einstein-Hilbert action (see refs. [274, 393]) √ Z −g S = d4x R + L ,(1) GR 2κ m where {xµ} the coordinates in the Einstein frame, g is the determinant of the µν µν metric tensor g , R = g Rµν the Ricci scalar, derived from the Ricci tensor Rµν, √ 8πG L = −gL the lagrangian density for the matter fields and κ = the m m c4 Einstein constant (with G Newton’s constant and c the speed of light). Generally, the action for a modified theory encapsulates the physical perception of the theory; it includes the Einstein-Hilbert action as a special case as well as terms that distance the theory from this. One can consider the following self-coherent and consistent cases of extending the Einstein-Hilbert action, by means of adding or altering terms to it, and thus transcending from General Relativity to a modified theory (see refs. [86, 289, 376]). 1. A simple class of metric theories of gravity: The addition of extra terms in the manner the typical matter fields are added. Such terms may include scalar fields, such as the Cosmological constant, the inflaton field, the chameleon field or the dilaton field, vector fields, such as the electromagnetic field or the Dirac spinors, and so on. Given the simplest case, we have the Cosmo- logical constant Λ and √ Z −g S = d4x (R − 2Λ) + L .(2) GR 2κ m However, the extra terms can be added in such a manner so as to interact with curvature. Given a scalar field, Φ = Φ(xµ), subjected to a potential V = V(Φ) and a Brans-Dicke function ω(Φ), the action becomes √ Z h −g ω(Φ) i S = d4x ΦR − (∂µΦ∂ Φ) − V(Φ) + L .(3) GR 2κ Φ µ m 2. The Lovelock class of metric theories of gravity: Lovelock attempted to generalise Einstein’s formulation, by including higher-order terms of curvature in the action. As a result, the following action can be taken as a general form, Z 4 p n SL = d x −g ∑ αnR + Lm ,(4) n 6 introduction

0 1 2 2 µν where R = 1, R = R the Ricci scalar, R = G = R − 4RµνR + κλµν RκλµνR the Gauss-Bonnet invariant, and so on, and αn the coupling constants [233]. A speciﬁc class of these theories is the Gauss-Bonnet theory, where the action contains up to the second-order terms of curvature, as an arbitrary function,

Z p 1 S = d4x −g R + f (G) + L .(5) GB 2κ m Other extensions of this are the major category of f (R) theories of gravity, where the Ricci scalar is substituted with an arbitrary function f (R), so the action turns to √ Z −g S = d4x f (R) + L .(6) f (R) 2κ m This category of models can be further extended by including the ﬁrst category, especially the case of scalar ﬁelds coupled to curvature terms [73, 119, 374, 375, 378]; other additions of metric-theories elements can be added as well, such as the Gauss-Bonnet invariant (see refs. [86, 289, 295, 376] for detailed accounts). Such models can be traced back to Starobinsky’s model [380], where second-order terms of curvature can, on the one hand eliminate major cosmological problems, such as the initial singularity, and on the other be accounted for quantum or string-theory corrections of General Relativity.

3. The Horndeski class of metric theories of gravity: Assuming the most general case of coupling between curvature and an auxiliary scalar ﬁeld in a four-dimensional space-time, that can produce second-order ﬁeld equations via a variational principle, we may obtain the following action [190] (see also refs. [65, 99, 127, 226] for recent developments),

Z 5 4 p 1 SH d x −g ∑ Li + Lm ,(7) 2κ i=2

where Li the curvature and scalar ﬁeld Lagrangian densities, with the following structure

L2 =G2(φ, X) , µν L3 =G3(φ, X)g ∇µ∇νφ ∂G 2 L =G (φ, X)R + 4 gµν∇ ∇ φ − gκµgλν∇ ∇ φ∇ ∇ φ and 4 4 ∂X µ ν κ λ µ ν µν L5 =G5(φ, X)G ∇µ∇νφ− 1 ∂G 3 − 5 gµν∇ ∇ φ + 2gαµgβνgγρ∇ ∇ φ∇ ∇ φ∇ ∇ φ− 6 ∂X µ ν µ β ν γ ρ α κµ λν αβ − 3g g g ∇κ∇λφ∇µ∇νφ∇α∇βφ , introduction 7

where Gµν the Einstein tensor (see next chapter), φ is an auxiliary scalar ρσ field, X = g ∇ρ∇σφ is kinetic term and Gi(φ, X) arbitrary functions of the latter two. This theory includes a majority of modified theories of gravity, such as the Brans-Dicke theory, the Quintessence theory, the Dilaton theories, the Chameleon and the covariant Galileons; it was considered to produce viable cosmological models that unified early- and late-time dynamics of the Universe (see inter alia refs. [120, 154, 226, 227, 266, 268, 285, 387]), but their relevance was recently questioned.

4. The bimetric theories of gravity: The introduction of a second metric that corresponds to the same points of space-time as the original metric. Hence, two Riemann-Christoffel curvature tensors exist and the action is written as a combination of the actions for the two different metrics √ Z p 4 −g − f SBi = d x R(g) + R( f ) + Lm ,(8) 2κg 2κ f

µν where f the determinant of the second metric tensor, f , κ f the Einstein constant in the second metric; each metric presents its very curvature and Planck mass, while the matter ﬁelds are considered to correspond to the original metric [349, 360]. Several viable cosmological models have been found in the context of this theory [14, 153, 228, 277], as well as possible explanations for dark matter [58].

5. The non-metric theories of gravity: The addition of geometric terms that have been (for some reason) “omitted” by Einstein in the original formulation of the theory can be continued into non-metric afﬁne manifolds, where the connection does not contain merely the Christoffel symbols, but also the torsion and non-metricity of space-time. Such theories are the Einstein- Cartan-Sciama-Kibble theory, where the Ricci tensor includes torsion terms, and the Einstein-Cartan-Weyl theory, also including the non-metricity,

Z p 1 S = d4x −g R + Rˆ + L ,(9) ECW 2κ m where ˆ α µ νρ α µ νρ µ νρ ν µρ R = M νρ M µα g − M µρ M να g + ∇µ(M νρ g ) − ∇µ(M νρ g ) ,

represents “scalar curvature” generated by torsion and non-metricity, where µ µ µ µ µ µκ λ µκ λ M αβ = −Kαβ + L αβ, and Kαβ = Sαβ − g gαλSκβ − g gβλSακ is the µ = µ contortion tensor (with Sαβ Γ [αβ] the torsion tensor), associated with the µ 1 µν altering in the slope of translated vectors, and L αβ = g Qναβ − Qανβ − 2 Qβνα the disformation tensor (where Qναβ = ∇νgαβ the non-metricity tensor), associated with the altering of the length of translated vectors [83, 181, 183, 185, 224, 276, 339, 361]. 8 introduction

Moving away from standard perception of curvature as gravity, we can also construct a theory of gravity where curvature is absent, but other geometric elements, e.g. torsion and non-metricity, are present. Such theories constitute the Teleparal- lel f (T) and f (Q) theories of gravity (see refs. [180, 183, 184] and more recently [76, 145, 156, 186, 265, 339]; main cosmological results can be found in refs. [77, 86, 144, 289] and [42, 77, 78, 146–148, 355] as examples). Among the many afore-mentioned variants, our focus lies with the latter cases, those of the f (R) theories, where standard curvature terms are present, but not in the simplified Einstein-Hilbert form. The purpose of this choice is that an abstract function f (R) allows for an intrinsic generation of cosmological phenomena, such as inflation and late-time accelerating expansion; the deSitter evolution of space- time can occur naturally from the careful choice of this function, so as it fits the behaviour of the specific cosmological era. The simple case of f (R) theories will be examined as for the generation of cosmic eras, by means of given Hubble rates for each of them. Furthermore, the combinations of f (R) theory with a scalar field, the Gauss-Bonnet invariant, the superfluidity of dark energy and dark matter, the Loop Quantum Gravity theory and the k-essence elements are examined, whereas additional terms are added in the action of eq. (6), so as to demonstrate their effects in the derivation of inflationary and late-time dynamics and the viability of these models. Two kinds of analytic tools are being used for the respective examinations. As long as the simple case and the scalar field and Gauss-Bonnet additional terms are considered, we reconstruct the corresponding f (R) model for different Hubble rates in a flat Friedmann-Lemaître-Robertson- Walker background and compare the results to current relativistic results and observational data; when the inflationary era is examined, we reconstruct the slow-roll indices and hence the spectral indices of scalar and tensor perturbations and compare these to the Planck and the BICEP2/Keck-Array data. As for the superfluid dark matter and dark energy, the Loop Quantum Gravity and the k- essence additions, specific dynamical models are obtained and their equilibrium points and corresponding stability are analysed. The remainder of this introduction is dedicated to a brief presentation of standard (relativistic) approaches of cosmic inflation and late-time accelerating expansion. In Chapter 2 and 3, we shall deal with the main theoretical frameworks of f (R) theory and the subsequent combinations to Gauss-Bonnet invariant; it i proved that ghost-free theories can exist whereas the combination of f (R) and Gauss-Bonnet invariant is assumed. For the sake of generality, the teleparallel counterpart of f (T) will be briefly presented. In Chapter 4, we shall examine the validity and viability of the standard f (R) theory, as given by the action of eq. (6), in describing the specific cosmic eras; aside from radiation-dominated and matter-dominated era, the deSitter and quasi-deSitter expansions will be examined, along with an exponential Hubble rate for early-time dynamics and a hyperbolic-tangent one for the late-time. Chapter 5 deals with the addition of the a scalar field and the Gauss-Bonnet invariant, in generating the early-time dynam- introduction 9 ics. Chapter 6 employs the theory of autonomous dynamical systems to explore the viability of the two-fluids cosmological model within the classical and the quantum framework, where the two fluids, namely dark energy and dark matter are coupled so that energy may be transferred from the one to the other; the case of Loop Quantum Gravity is also presented and the subsequent instabilities or divergences from the classical case are discussed. Finally, in Chapter 7, an autonomous dynamical model is presented that encapsulates the combination of f (R) theory with k-essence; the equilibrium points and consequent stability (or instability) are illuminated, along with their consequences for the validity of the model. 10 introduction

1.1 the standard model and the cosmological eras

The current beliefs about the large-scale structure and evolution of the Universe can be summarised in the Cosmological Principle and the Λ-CDM FLRW model. The formerresults from the Copernican principle, that people on Earth are not privilidged observers, and the “stylised fact” that the observable universe seems homogeneous and isotropic in the large-scale at all directions; as e result, the Uni- verse is homogeneous and isotropic, notably it allows all observers to observe the same thing at any direction of observation1. The latter can also be stated as the Standard Cosmological Model; what it encloses is that the space-time is ﬂat and spatially expanding according to the Friedmann-Lemaître-Robertson-Walker metric, containing aside from typical luminous matter (relativistic or non-relativistic), cold dark matter and a Cosmological constant, Λ. According to observations (see refs. [243, 395] as standard textbooks and [79, 389] as more recent ones; [36, 50, 187, 259, 347, 408, 412] bare constraints from Type Ia Supernovae, galactic clusters, etc., while refs. [6, 8–10, 13, 383] contain results from the Planck collaboration as well as the BICEP2/KeckArray) and as stated before, the proportions of typical luminous matter, cold dark matter and an effective ﬂuid representing the Cosmo- logical constant -usually referred to as dark energy- are ∼ 4%, ∼ 27% and ∼ 76% respectively. This standard model goes along with a hronology of the Universe, or rather with an exact sequence of cosmological eras that are supposed or porved to have occured from the “beginning” of time. These can be summarised as follows [389, 395]:

1. t = 0 sec: the “Big Bang”. The initiation of cosmic time as described by a gauge invariant theory of space-time, such as the General Reltivity.

2. Until ∼ 10−12 sec : the Very Early Universe. The first pico-second of cosmic time. It can be further divided to: i The Planck epoch: < 10−43 sec. The scale of the 3-d space is minimal; the radiation temperature of the universe is believed to lie at about 1032 K or more. In so high energies, the currently understood laws of physics may not apply and the General Relativity should fail to explain; it is believed that the four fundamental interactions were combined to one unified interaction at this point, breaking to gravitational and electrostrong interactions by the end. ii The Grand Unification epoch: < 10−36 sec. The scale of the 3-d space remains small; the radiation temperature falls but remains higher than

1 Recent observational data from the Planck collaboration find a small but statistically significant anisotropy, both in the background temperature fluctuations and in the primordial perturbations. These results point either to drop the Cosmological Principle and hence the Standrard Cosmolog- ical Model, or to extend the relativistic theory of gravity, even to abandon it in the favour of an (unknown yet) alternative. 1.1 the standard model and the cosmological eras 11

1029 K. The electromagnetic, the weak nuclear and the strong nuclear interaction remains unified throughout this era. iii The Cosmic Inflation: from 10−36 until 10−32 sec. The scale of the 3-d space expands rapidly by a factor of 1026; the radiation temperature falls from 1028 K to 1022 K, a situation known as supercooling. The strong interaction becomes distinct from the electroweak. iv The Reheating epoch: < 10−12 sec. The spatial universe remains loosely at the same size; the temperature is risen to its original values (∼ 1027 K). 3. Until ∼ 377000 yrs: the Early Universe. Throughout this era, the spatial components of the Universe grow at steady rates, much smaller than the inflationary. At the beginning of this era, various kinds of subatomic particles are formed; both matter and antimatter particles are formed at almost equal amounts, so the majority of them annihilates, leaving a small excess of matter due to the CP violation. Later, neutrinos decouple and afterwards the primordial nucleosynthesis occurs, with Hydrogen nuclei (free protons) merging with free neutrons into Deuterium nuclei, Helium nuclei, Lithium nuclei, etc. Following the nucleosynthesis, the Universe is filled with a dense and opaque plasma consisting of the afore-mentioned nuclei and free electrons (and other subatomic particles, such as muons, neutrinos, etc.), where radiation being unable to emerge. Gradually, matter dominates radiation. At the end of this era, the temperature and density of the Universe become low enough for atoms to form out of free nuclei and electrons; the atoms quickly reach their ground states by emitting photons. Subsequently, the plasma gradually disappears and the Universe becomes transparent, so that radiation (mainly the emitted photons) are free to propagate and form the Cosmic Microwave Background radiation. The division of this era into smaller ones follows as: i The Quark epoch: from 10−12 until 10−6 sec. The radiation temperature of the universe is higher than 1012 K. With the fundamental interactions separated but with too high energies, the quarks cannot yet coalesce into hadrons; unable to escape their coupling with gluons, they form a dense quark-gluon plasma. Though this stage has never been observed, energies such as this can be reached in the Large Hadron Collider, where this plasma is said to be observable. ii The Hadron epoch: from 10−6 until 1 sec. The radiation temperature lies higher than 1010 K. In these levels of energy, quarks are bound into hadrons (protons, neutrons, pions, etc.). The baryon asymmetry occurs, due to the CP violation, and all anti-hadrons are supposed to be eliminated. iii The Neutrino decoupling: 1 sec. The radiation temperature lies at about 1010 K; the “observable universe” reaches a radius of approxi- 12 introduction

mately 10 light years. At this point the neutrinos cease to interact with baryonic matter and are capable to propagate. iv The Lepton epoch: from 1 until 10 sec. The radiation temperature falls close to 109 K. Leptons and anti-leptons remain in thermal equilibrium. v The Nucleosynthesis epoch: from 10 until 103 sec. The radiation temperature falls to ∼ 107 K; the “observable universe” grows up to 300 light-years; the baryonic matter density lowers to ∼ 4 g/m3. Protons and neutrons are bound into atoms, mostly Hydrogen and Helium-4. Small amounts of Deuterium, Helium-3, Lithium-7 and some heavier elements are also synthesised. vi The Photon epoch: from 103 sec till 377000 yrs. The radiation temperature falls from 109 K to 4000 K. The Universe consists of the dense and opaque plasma, since atoms cannot yet form. Radiation is coupled in this plasma, unable to propagate free. vii Recombination: prior to 380000 yrs. The radiation temperature is at ∼ 4000 K; the Hubble radius reaches 42 106 light-years; the baryonic matter density has dropped to 5 108 atoms per cubic meter (approximately 109 times higher than current values). During this era, free electrons and atomic nuclei coalesce into neutral atoms. Photons escape thermal equilibrium with matter, as the latter becomes cooler and ex- cited atoms reach their ground state. Gradually, the Universe becomes transparent to photons and the latter propagate freely, forming the Cosmic Microwave Background radiation.

4. Between 380 Kyrs and 150 Myrs: the Dark Ages. In this stage of cosmic evolution, the radiation temperature falls from 4000 K to 60 K. The freely propagating CMB photons red-shift to infrared, so the Universe is no longer visible in general; due to this, for the ﬁrst time we can assign values to cosmic redshift, z, initiating at 1100 and ﬁnishing at 20.

5. Between 150 Myrs to 1 Gyr: Formation and Evolution of Structure. The radiation temperature drops from 60 K to 19 K; the cosmic redshift drops from 20 to 6. Stars, galaxies and galactic clusters begin to form. It is theorised that galaxies existed from ∼ 380 Kyrs, but only fully coalesced into “proto- clusters” by the end of this era; galaxy clusters appear later, at ∼ 3 Gyrs, while superclusters at ∼ 5 Gyrs. It is also theorised that stars were formed before galaxies (the “bottom-up” approach), although minihalos and gas clouds in the size of globular clusters existed prior to the formation of stars, during this era. i Early generations of stars: from 200 Myrs to 500 Myrs. The ﬁrst generations of stars and subsequently those of galaxies form and early large structures emerge, drawn to the foam-like dark matter ﬁlaments; 1.1 the standard model and the cosmological eras 13

dark matter has already evolved to large-scale structures prior to typical, due to its non-coupling with radiation. The earliest generations of stars are supposed to be huge (at about 100 to 300 solar masses), non- metallic and short-lived (in comparison to contemporary stars), hence they blow up to supernovae rather fast, creating most of the heavy elements of present-day Universe. ii Galaxies, dwarf galaxies and quasars: from 250 Myrs to 900 Myrs. After the ﬁrst generation of stars, dwarf galaxies and probably quasars emerge, emitting high energy photons. iii Reionization era: about 700 Myrs until 1 Gyr. The emission of high energy photons causes the reionization of atoms.

6. ∼ 13.8 Gyrs: Present time. The radiation temperature has reached 2.726 K; the Hubble radius is measured ∼ 93 Giga-light-years, containing a volume of 4 1080 m3 and mass equal to ∼ 4.5 1051 kg; the density of the total energy and mass is approximately 9.9 10−24 g/m3, equivalent to 6 protons per cubic meter. As stated, this model is based on the Friedmann-Lemaître-Roberton-Walker (FLRW) metric of a ﬂat homogeneously-expanding space-time, in the form

3 2 ds2 = −dt2 + a(t)2 ∑ dxi ,(10) i=1 where a(t) is the scale factor of the Universe2. We can denote the Hubble rate as the expansion/contraction variable of the 3-d space, deﬁned as the rate of change of the scale factor, a˙ H = .(11) a 3 We can also deﬁne the deceleration parameter as

H˙ aa¨ q = − 1 + = − ,(12) H2 a˙2 measuring the acceleration in the expansion of the 3-d space4.

2 Originally, the Robertson-Walker metric contained the case of curved space-time and was written in spherically symmetric coordinates as dr2 ds2 = −dt2 + a(t)2 + r2dθ2 + r2 sin2 θdφ2 , 1 − kr2 but eventually, k = 0 was chosen since it matches the present-time observations; usually, k = −1 denotes a closed universe, destined to collapse in a “Big Crunch”, while k = 1 stands for an open universe, reaching an eternal highly-accelerating expansion. 3 During the whole dissertation, dots over variables will indicate derivatives with respect to the proper time 4 Following ref. [392], we can also deﬁne the jerk parameter HH¨ j = ,(13) H˙ 2 14 introduction

The curvature of the ﬂat FLRW space-time can be easily extracted by means of temporal derivatives of the metric. The Ricci tensor is fully diagonal; its temporal and spatial components yield

a¨ R = −3H˙ + 6H2 = −3 and R = R = R = H˙ + H2 = aa¨ + 2a˙2 .(15) tt a xx yy zz Contracting, we obtain the Ricci scalar, that is

a¨ a˙ 2 R = 6H˙ + 12H2 = 6 + .(16) a a

The Universe is dominated by an ideal ﬂuid, of mass-energy density ρ and pressure P, that are interlinked via an equation of state; assuming the ﬂuid to be barotropic, the equation of state can be written as

P = wρ ,(17)

P where w = is the barotropic index. The energy-momentum tensor of such a ρ ﬂuid is diagonal, with the mass-energy density in the temporal component and the pressure in the spatial ones,

Tµν = diag(−ρ, P, P, P) .(18)

Substituting these to the Einstein equations, and preserving only the temporal diagonal and the (contracted) spatial diagonal elements, we obtain the Friedmann equation, a˙ 2 κ H2 = = ρ ,(19) a 3 and the Landau-Raychaudhuri equation,

a¨ a˙ 2 κ H˙ = + = − (ρ + P) .(20) a a 2 From the law for the conservation of energy and momentum, we obtain the continuity equation, ρ˙ + 3H (ρ + 3P) = 0 . (21) Either the Friedmann or the Raychaudhuri equation, the continuity equation and the equation of state constitute a complete system of equations describing the evolution of matter and space in the large-scale Universe.

measuring the rate of change of the acceleration in the expansion, and the snap parameter ... H2 H s = ,(14) H˙ 3 measuring the acceleration of the acceleration in the expansion. 1.1 the standard model and the cosmological eras 15

Given that P = wρ, eqs. (20) and (21) yield the following solutions, proposing that w 6= −1: ﬁrstly, the mass-energy density is linked to the scale factor as ρ a −3(1+w) = ,(22) ρ0 a0 where ρ0 and a0 are the values of mass-energy density and scale factor at a given moment of cosmic time (e.g. the present-time); secondly, substituting this to eq. (19) or (20), the scale factor is generally given as

2 3(1+w) a = a0t (23) and consequently, the Hubble rate is 2 H = ;(24) 3(1 + w)t ﬁnally, substituting back to the mass-energy density, we have

−2 ρ = ρ0t .(25) We can easily see that the expansion of the Universe, as it reflects on the scale factor and the Hubble rate, depends both on time and on the barotropic index (essentially on the type of fluid). However, the mass-energy density drops over time as t−2, no matter the exact content of the Universe. More specifically, we can assign values to the barotropic index and examine special case, where the cosmic fluid behaves as relativistic matter (radiation), non- relativistic matter (dust), stiff matter, or even as an effective fluid for Cosmological constant or spatial curvature. The afore-mentioned five cases may represent a cosmological era, given that a specific element dominates the Universe at each stage of each development. 1. Late-time accelerating expansion. Given that w = −1, the cosmic fluid has negative pressure, equal and opposite to its mass-energy density; this can represent the effects of Cosmological constant. The afore-mentioned solutions for the mass-energy density and the scale factor are not consistent with this case, due to their derivation. It is proved, however, from eq. (21) that the mass-energy density is constant

ρ = ρ0 , while the scale factor can be found to rise exponentially as √ κ ρ0 a = a0e 3 ,

where κρ0 = Λ can be viewed as the Cosmological constant; the Hubble rate is also constant and equal to the effective Cosmological constant, as r rκ Λ H = ρ = . 3 0 3 16 introduction Hubble rate Hubble rate

Mass-Energy Density Mass-Energy Density Hubble rate Hubble rate

Mass-Energy Density Mass-Energy Density

Figure 1: The phase space of eqs. (20) and (21) for different values of w, corresponding to different types of fluid. Top-left has w ≥ 0, reflecting all typical matter fields, such as relativistic matter (radiation) or non-relativistic matter (pressure- 1 less dust); top-right has w = − , standing for spatial curvature as an effective 3 fluid; bottom-left has w = −1, that represents the canonical Cosmolgical constant; bottom-right has w < −1, that represents the phantom or qintessential Cosmological constant. The latter cases correspond to the late-time accelerating expansion, while the first one can stand for classical cosmic eras, such as the matter-dominated and the radiation-dominated eras; the second would be a good (tjough mistaken) approximation for the inflationary era. Blue arrows stand for the vector field, red curves for possible trajectories, and the dashed black curves represent the central manifolds of the equilibrium in (0, 0). Both the mass-energy density and the Hubble rate are rescaled.

This result is reached also from the deSitter solution of General Relativity, r Λ for an empty Universe expanding exponentially with a factor . This 3 evolution of the Universe corresponds to the present-time evolution (for redshifts less than 0.4), where an unknown component of the matter ﬁelds (usually called “dark energy”) or some unknown geometric feature (that 1.1 the standard model and the cosmological eras 17

the modiﬁed theories of gravity try to supply) drives the universe to accelerating expansion, as indicated by the Type Ia Supernovae data.

2. Matter-dominated era. Given w = 0, the cosmic fluid has zero pressure and corresponds to pressureless dust; this can represent the case of uniform cold non-relativistic matter that dominated the Universe prior to the “dark-energy”-dominated era (for redshifts between 3600 and 0.4). In this era, the cold baryonic and leptonic matter dominated the hot and massless (or nearly massless) relativistic particles, such as photons, neutrinos, etc., resulting to a decelerating expansion of space. Notably, the scale factor becomes 2 a = a0t 3 and the deceleration parameter is easily found to be 3 q = , 4 which is always greater than 0. 1 3. Radiation-dominated era. Given w = , the cosmic fluid has positive pres- 3 sure, smaller than its mass-energy density; this corresponds to a fluid composed of relativistic particles and is able to describe the case where photons, neutrinos, and other such hot and massless (or nearly massless) relativistic particles dominated the evolution of the universe. This era is accompanied with an even greater decelerating expansion of space and corresponds to redshifts greater that 3600, before the matter-radiation equivalence moment, when the effects of matter on expansion balanced out and surpassed the effects of radiation. The scale factor becomes

1 a = a0t 2

and the deceleration parameters is found to be

q = 1 .

4. Emergence of spatial curvature. As we stated, it is generally perceived that the Universe is spatially flat, at least in later stages; however, it would be possible for curvature to emerge in previous stages of evolution, dominating 1 matter and radiation. Given w = − , this case is exhausted within the flat 3 FLRW model, since curvature can be expressed as an effective fluid with negative pressure. In this case, the scale factor is

a = a0t

and the Hubble rate, 1 H = . t 18 introduction

Th deceleration parameter is easily found to be zero,

q = 0 .

Consequently, if the Universe was at any time dominated by the effects of non-zero spatial curvature, this would imply a constant expansion with a linear scale factor.

5. Stiff matter as inflation. The earliest stage that can be examined with the flat FLRW model is that of inflation; in this era, both matter and radiation are dominated by an unknown field (e.g. the inflaton) that drives the Universe to accelerating spatial expansion with a constant Hubble rate. Zel’dovich theorised such an early stage of cosmic evolution, where matter fields are composed of a cold gas of baryons behaving like stiff matte, while Bose- Einstein condensates are also possible cases (see refs. [101, 103] for example). This case corresponds to w = 1, essentially a fluid with pressure equal to the mass-energy density and the speed of sound reaching the speed of light. Such an era of the Universe yields a decelerating expansion, with scale factor 2 a = a0t 3 and deceleration parameter 2 − 2 q = t 3 ; 3 it should be noted that th deceleration parameter drops over time, but never reverses to negative values (so acceleration is made possible). With current state of inflationary dynamics (briefly examined in the next section), the decelerating expansion of space is not consistent, but a constant acceleration similar to the emerging spatial curvature, or the deSitter expansion is expected; as a result, Zel’dovich’s argument of a stiff-matter-dominated era is dropped.

1.2 cosmic inflation and scalar fields

The inﬂationary era of cosmic evolution is essential to the FLRW Λ-CDM model, since it allows for the major Big Bang puzzles to be resolved within the model, without the abolishment of neither the initial singularity, nor the Cosmological principle. In fact, the inﬂationary scenario was proposed [172, 173, 212, 247] as a result of the Higgs boson, as a phase transition covering the early stage of cosmic evolution allowing for the braking of symmetry and the attribution of mass to all particles due to the Higgs mechanism. However, it was immediately used [16, 171, 250, 252, 381, 399] to deal with the following puzzles of the Big Bang model (for further details see refs. [51, 79, 160, 220, 244, 254, 255, 279]). 1.2 cosmic inflation and scalar fields 19

1. The Cauchy problem of the Universe. The conventional Big Bang model requires a fine-tuned set of initial conditions along with the Standard Cos- mological model, so that the evolution of the Universe can proceed and reach the Universe observed today, rather than a completely different one. Specifically, if we consider a 3-d spatial slice of constant time and define positions and velocities of all matter particles at that specific moment, the system should evolve according to the laws of gravity and fluid dynamics, in short according to the equations (19), (20) and (21), along with an equation of state; this can generally lead to many diverging evolutions, depending strongly on the initial conditions, only one of which reflects the Universe we line in. i Initial homogeneity: According to observations, small inhomogeneities existed in the past stages of the universe (during the Recombination era, since they are observed in the CMB) and were probably even smaller in the early universe (since inhomogeneities are gravitationally unstable and should grow over time). Therefore, a question rises as for the initial homogeneity of the Universe that somehow breaks and small inhomogeneities appear, that will later on lead to the formation of structure. ii Initial velocities: Demanding that the Universe remains homogeneous at late times, we require that initial velocities of the cosmic fluid were really small. However, if this is true, the Universe cannot expand as we know (or assume) it did; on the contrary, it recollapses within a fraction of a second. Furthermore, if the initial velocities are sufficiently large, the Universe expands too rapidly and ends up to a flat empty space- time (Minkowski-like) in very short time. Consequently, it seems that all initial velocities of particles in the cosmic fluid need to be fine-tuned, so that either catastrophe is avoided. The physical and philosophical problem behind the Cauchy problem is that no fundamental reason for this fine-tuning exist. Theoretically, any 3-d spatial slice, regardless of initial conditions, should evolve forward in time and produce the Universe we observe today, due to the fact that the very same laws of physics apply to all such spatial slices at any moment time. There- fore, we have no reason to assume that initial conditions were chosen, imposed or accidentally set so that the accounted evolution took place and the present-time Universe was established.

2. The Horizon problem. Let us consider a photon or some other particle propagating at the speed of light across the FLRW ﬂat space-time. The comoving horizon of such a particle is deﬁned as the causal horizon, or as the maxi- mum distance the light can travel between two moments of cosmic time, t1 20 introduction

Z dt and t . Considering the conformal time, τ = , the causal horizon is 2 a(t) given as Z t2 dt dp = τ2 − τ1 = ,(26) t1 a(t) and the physical size of this horizon as

d˜p = a(t)dp .(27)

The causal horizon can be perceived as the fraction of space-time (of the Universe) that is in causal contact, essentially whose smaller fraction have contacted via the speed of light between the two moments of cosmic time, t1 and t2, so each of them “knows” all others; this is referred as the Hubble radius. We may always consider the initial moment of cosmic time as the beginning of time, t1 = 0, and the ﬁnal moment as any miscellaneous moment, t2 = t; as a result, the comoving horizon for a photon propagating since the initial singularity can be taken as

Z t dt0 Z a da0 Z a d ln a0 = = = = dp τ 0 02 0 ,(28) 0 a(t ) 0 Ha 0 a H

that is the fraction of the Universe that has been causally connected from the beginning of cosmic time until now. Given that the Universe is dominated by a fluid, with equation of state P = wρ, the comoving horizon, τ increases with time as 1 (1+3w) τ ∝ a 2 ,(29) according to barotropic index of the fluid; notably, the moment of matter- radiation equivalence, when (1 + 3w) changes from 2 (radiation-dominated) to 1 (matter-dominated), plays a very crucial role, as τ ∝ a before recombi- 1 nation and τ ∝ a 2 afterwards. We come to understand that the comoving horizon grows monotonically over time, hence comoving scales that enter the horizon in present-time must have been far outside the horizon during the recombination and the photons decoupling (the time of the CMB). The question arising is how the extreme homogeneity of the Universe (in all its different fractions) holds true for very early stages (prior to the recombination) when the corresponding fractions of the Universe are non-causally connected and independent. From the side of the inflationary scenario, this can be resolved as follows: a brief period of vast expansion of the 3-d space, allows for the Hubble radius to shrink and then to re-expand, in other words for fraction of the Universe that were in causal contact to move far away from each other and become independent. These fractions of the Universe contain scales relevant to cosmological observations today that remained larger than the causal horizon until a ∼ 10−5 and re-entered the Hubble radius later, as the Universe expanded, while already causal. Their causal 1.2 cosmic inflation and scalar fields 21

Figure 2: The solution to the Horizon problem. The comoving Hubble radius (aH)−1 shrinks during inﬂation and expands afterwards.

connection has been established prior to inﬂation and kept so far, hence no difference in their homogeneity or isotropy can be observed.

3. The Flatness problem. Deﬁning the critical density of the Universe as ρcrit = 3κH2, from eq. (19), that contains all matter ﬁelds contains of a relativistic FLRW space-time, we may consider the curvature parameter as

ρ − ρ ρ − 3κH2 = crit = Ωk 2 ,(30) ρcrit 3κH which is essentially a difference between the average potential energy and the average kinetic energy of a region of the Universe. As long as the space- time is ﬂat, as we have supposed, then eq. (19) holds true in this form and therefore Ωk = 0; however, this should not just be the case. Taking into account that the mass-energy density of the Universe is not time-independent, but rather ρ = ρ(α), we should rewrite

k ρ(a) Ω (a) = − = 1 − ,(31) k (aH)2 3κH(a)2 so the curvature parameter is also time-dependent. Differentiating and using the continuity equation (21), we may obtain the differential equation dΩ k = −(1 + 3w)Ω (1 − Ω ) ,(32) d(ln a) k k

which has an unstable equilibrium point at Ωk = 0, as long as the strong energy condition is satisﬁed

1 + 3w > 0 . 22 introduction

This can be supported from theoretical results from the Big Bang Nucle- ρ − BBN ≤ O( −16) osynthesis era, yielding 1 2 10 , the Grand Uniﬁed The- 3κHBBN ρ − GUT ≤ O( −55) ory era, yielding 1 2 10 , and the Planck epoch, yielding 3κHGUT ρ − Planck ≤ O( −16) 1 2 10 . The question arising is how the spatial curva- 3κHPlanck ture of the Universe is chosen, imposed or accidentally set equal to zero, while (similar to the Initial Conditions problem) no physical mechanism leads to it.

4. The Cosmological Perturbations problem. The present-time Universe is filled with small- and large-scale structure that has been formed due to the gravitational instability (or Jeans instability), which -as we noted- allows for small inhomogeneities to expand and therefore, small perturbations in the mass-energy density and curvature of the Universe are considered to have grown in size and resulted to the structure formation. However, the extreme homogeneity of the Universe in the very early stages, also noted in the Ini- tial Conditions problem, is not capable to offer such perturbations; neither some other physical mechanism exists within the Standard Cosmological model. As a result, a question arises as of how the initial cosmological perturbations raised. A period of inflation, that would expand the 3-d space so as to increase existing quantum fluctuations (of very small scale) into large-scale classical fluctuations, is a convenient answer.

5. High Energy Physics problems. Aside from the classical-physics problems described so far, other problems also exist in the Standard Cosmological model, especially is put side by side with the Standard model for Elemen- tary Particles. Essentially, situations that require high energy conditions are attributed to the early stages of the Universe, but are absent in the later as the Universe cools down and “becomes classic”; the main such problem is the Magnetic Monopoles problem. The Magnetic Monopoles or Exotic Relics problem. The high energies at the very early Universe suggest the existence of large number of very heavy, stable magnetic monopoles, as proposed by the Grand Uniﬁed Theories. These magnetic monopoles are heavy and stable particles representing the “magnetic charge” (in the same manner we have “electric charge” in classical theory), therefore they should be existing until today after their formation at the very early Universe, yet they have not been observed in nature. According to current literature [79, 160, 244, 395], a period of inﬂation that occurs below the temperature where magnetic monopoles can be produced may offer a possible resolution of this problem, since the monopoles would be separated from each other due to the expansion of 3-d space and their observed density 1.2 cosmic inflation and scalar fields 23

would be decreased by many orders of magnitude, practically making them unobservable at present-time

To deal with inﬂation, one needs speciﬁc conditions. These can follow from the following equivalent statements [51, 160, 220, 243, 244, 254, 255, 395]:

• A decreasing comoving horizon: the Hubble radius must decrease -as the solution to the Horizon problem dictates; thus, the fundamental definition for inflation can follow d 1 < 0 . (33) dt aH This is directly related to the generation of cosmological fluctuations.

• An accelerated expansion: since the Hubble radius must shirnk, we may easily obtain the following relation

d2a > 0 , (34) dt2 that implies a rapidly increasing Hubble rate. From here, we may deﬁne an acceleration index a H˙ d ln H ε = − = − ,(35) H2 dN where N is the e-foldings number deﬁned as

Z t f in N = Hdt ,(36) tin

−36 with tin the initial and t f in the final moment of inflation (about tin = 10 sec and t = 10−32sec, so that the inflation should last for more than 60 e-folds). Generally, accelerating expansion corresponds to

ε < 1 .

• The pressure of the matter ﬁelds is negative: the accelerated expansion can be triggered only by matter ﬁelds with extremely negative pressure; from eq. (20) and supposing a¨ > 0, we easily see that

1 P < − ρ ,(37) 3 which requires a barotrobic ﬂuid (real or effective) much more un-stiff than the effective ﬂuid representing spatial curvature.

In order for this to become consistent with some physical mechanism, many ideas were offered (see, for instance refs. [242, 245, 247, 280] or [79, 160, 244, 254, 255, 258]), usually tied with a scalar field, that would slow-roll on its potential and trigger inflation; this scalar field was usually associated with some quantum 24 introduction

correction to gravity, or some emergent quantum field that would be active during the early stages of the Universe [173, 212, 381, 399]. Following [51], we present a simple scalar field model that can slow-roll and the standard inflation, as well as the subsequent reheating; similar examples can be found in refs. [79, 160, 243, 279]. Such models are highly appealing in the literature [242, 245], even among the modified gravity theories (see [289]). Later models, such as eternal chaotic inflation (see refs. [248, 249, 251, 253] among many) move away from this notion, since inflation may never actually stop. We shall assume the action of eq. (3) with ω(φ) = 1, where a scalar field, φ, is minimally coupled to gravity. Notably, such actions can be split to the original Einstein-Hilbert action plus an action fro the scalar field -the latter being similar to the actions used in quantum field theory; furthermore, typical matter fields can be considered zero and the scalar field can be viewd as an effective fluid dominating the 3-d space. The potential V(φ) describes the self-interactions of the scalar field and needs to contain a global minimum, where the scalar field can oscillate, and an almost flat plateau, where the scalar field can slow-roll (see fig. 3); the latter part corresponds to the slow-roll condition for a Starobinsky inflation (see ref. [380]), while the former corresponds to the post-inflation reheating phase, where the Einstein gravity is dominant.

Figure 3: A typical potential for the scalar ﬁeld, φ. The plateau on the right corresponds to the slow-roll inﬂation, while the minimum on the left to the reheating.

In the absence of typical matter fields, the energy-momentum tensor for the scalar field is ( ) 1 T φ = ∂ φ∂ φ − g gκλ∂ φ∂ φ + V(φ) ,(38) µν µ ν µν 2 κ λ 1.2 cosmic inflation and scalar fields 25 from where the following energy density and pressure can be derived, expressing the effective fluid in a Friedmann-Lemaître-Robertson-Walker space-time,

1 1 ρ = φ˙ 2 + V(φ) and P = φ˙ 2 − V(φ) .(39) φ 2 φ 2 Of course, the resulting effective equation of state is

1 2 Pφ φ˙ − V(φ) w = = 2 ,(40) φ ρ 1 ˙ 2 φ 2 φ + V(φ) 1 which can lead to negative pressure and accelerated expansion w < − , e f f 3 proposing that the potential surpasses the kinetic term. The dynamics of the (homogeneous) scalar ﬁeld and the FLRW geometry can be expressed by means of a continuity equation dV φ¨ + 3Hφ˙ + = 0 , (41) dφ derived from eq. (21), and a Friedmann equation

1 1 H2 = φ˙ 2 + V(φ) ,(42) 6 3 derived from eq. (19). For large values of the potential (relatively to the kinetic term), the scalar field experiences significant Hubble friction from the term Hφ˙, which is responsible for the slow-roll. H˙ Having defined the acceleration parameter, ε = − , we may proceed by ap- H2 plying the slow-roll condition,

V(φ) φ˙ 2 ,(43) and deﬁning two parameters to measure the shape of the inﬂationary potential,

2 2 dV d V M dφ 2 dφ2 e (φ) = Planck and η (φ) = M2 ,(44) V 2 V(φ) V Planck V(φ) that are called the potential slow-roll indices; ensuring that e , |η| < 1, we ensure that the fractional change of e per e-fold is small; ensuring that e , |η| 1, we enter the slow-roll regime, where

1 1 dV H2 ' V(φ) ' const. and φ˙ ' − , 3 3H dφ therefore the space-time is approximately deSitter and evolves according to

1 V(φ)t a ∼ e 3 . 26 introduction

The inflation ends when the slow-roll conditions are violated, specifically when eV ' 1. From this, we can calculate its duration in terms of the e-foldings number, since t a f in Z f in Z φf in H N = ln = Hdt = dφ , ain tin φin φ˙ which is proved to be Z φin dφ N(φ) ' p .(45) φf in 2eV (φ) In order for the Horizon and the Flatness problems to be resolved as described, it is required that the total number of e-foldings exceeds about 60. The precise number depends on the enrgy scale of inflation and on the details of reheating afterwards; the fluctuations observed in the CMB seem to be created at about 40 to 60 e-folds before the end of inflation, so close to its initiation. As for the reheating, it is an outcome of the scalar field oscillations around the minimum of the potential, oscillations that follow the slow-roll of the field on the plateau of the potential. During this phase of coherent oscillations, the scalar fields behaves like pressureless matter (dust). However, the coupling of the scalar field to other (massive) particles leads to a decay of the former’s energy, slowly converting it to into Standard Cosmological model degrees of freedom. As a result, the supercooling that was brought duw to the inflation is recovers, the temperature of the universe returns to its pre-inflation levels, and the hot Big Bang commences. This is exactly the importance -and the necessity- of reheating; if the inflation is to take place as described, the Universe is supercooled prior to the formation of matter particles (baryons and leptons), so it cannot lead to the present-day observed Universe; in order for both the inflationary scenario to hold and the hot Big Bang to continue as described, producing the particles we observe today, a phase of reheating after the inflation is necessary.

1.3 accelerating expansion and cosmological constant

Following the observations of type Ia Supernovae -white dwarfs that exceed their stability limit and explode to supernovae- many evidence exist that the expansion of the Universe in its latest phase is accelerating, rather than decelerating as the matter-domination Friedmann model suggests. The redshift of these supernovae indicated that the scale factor at the time of their explosion should be a a(t) = 0 ,(46) 1 + z(t)

where a0 the current value of the scale factor, that we can set to unity; this results to the Universe taking longer time to expand from its given state (at the time of the explosion) until its current one, given a realistic assumption for its mass-energy density (about 20 ∼ 25 %) and a ﬂat FLRW background. In fact, the only way the resulted acceleration would be possible was with the re-introduction 1.3 accelerating expansion and cosmological constant 27

of a constant in the small redshifts, corresponding to the late-time evolution of the Universe, that would break the matter-dominated era and cause an accelerating expansion5. Observations from Baryon Acoustic Oscillations, originating in the matter decoupling approximately 380000 years after the Big Bang, and in the measurements of the mass functions of galaxy clusters, conﬁrmed the afore- mentioned conclusion, that in fact the Universe is expanding at an accelerating rate. This present phase of the Universe rapid expansion is found to correspond almost exactly to a deSitter expansion, described by a constant Hubble rate

H = H0 = const. , and hence resulting to an exponential scale factor, a(t) ∼ eH0t. This exponential expansion of the 3-d space could be explained if the Universe was either dominated by some unknown field, represented by the Cosmological constant Λ, or by a fluid with negative pressure, whose equation of state is P = wρ , 1 where w < − [52, 68, 130, 395]. Several ideas has been proposed as for why this 3 unknown component behaves like that, what could it actually be, and why did it appear so late in the cosmic evolution. Some ideas can be encapsulated in the classical theory of General Relativity, i.e. the introduction of an arbitrary scalar field, while others can be formulated by extending the classical theory in the sense of the metric or non-metric theories of gravity presented above (refs. [408, 412] set boundaries on the arbitrary scalar fields, while ref. [198] among others applies constraints on the modified theories of gravity, all based on observational data). However, none of these cases was proved to have some physical meaning, that is to correspond to a specific physical mechanism represented by the arbitrary scalar field or by the additional geometric terms in the action. Henceforth, we shall briefly present the main ideas formulated in order to describe the accelerating expansion, namely those that were proposed as both mathematical concepts and physical mechanisms behind the late-time acceleration and were not proved as successful.

1.3.1 Canonical Dark Energy

Dark energy is a an unknown form of energy that is hypothetised to permeate the 3-d space, that can be formulated as an effective ﬂuid with barotropic index

5 When Einstein published his theory of gravity, the General Relativity, in 1916, he introduced a constant Λ in the equations in order to keep the Universe at a steady state, but without any other mathematical or physical reasoning; this constant was later removed and deemed “his greatest mistake”, due to the observations by Edwin Hubble, that the galaxies are distancing themselves from each other, and hence the Universe seems not to be steady but rather to expand. The re- introduction of this constant in the late 90’s was in order to explain the current phase of expansion, that was found by Perlmutter, Riess and Schmidt to be greater than expected. 28 introduction

we f f = −1, that is with negative effective pressure. This is the most accepted and convenient hypothesis, since it can be easily parametrized under the Cosmo- logical constant. Rewriting the Friedmann equation (see refs. [130, 395] as well as [17, 36, 187, 259, 412] for details), using the fact that the mass-energy density is disputed between the afore-mentioned components of the Universe, namely between radiation with mass-energy density

a −4 ρrad = ρrad(0) , a0 dust -representing both (typical) baryon matter and cold dark matter- with mass- energy density a −3 ρb + ρcdm = rb(0) + rcdm(0) , a0 curvature with effective mass-energy density

a −2 ρk = ρk(0) , a0 and dark energy with effective mass-energy density

a −3(1+wDE) ρDE = ρDE(0) ; a0 so eq. (19) may be written as q −4 −3 −2 −3(1+w ) H(a) = H0 Ωrada + (Ωb + Ωcdm) a + Ωka + ΩDEa DE ,(47)

rκ where H = H(a ) = ρ the present value of the Hubble rate corresponding 0 0 3 crit to the critical mass-energy density of the Universe, ρcrit, and

ρrad(0) ρb(0) ρcdm(0) Ωrad = , Ωb = , Ωcdm = , ρcrit ρcrit ρcrit ρk(0) ρDE(0) Ωk = and ΩDE = , ρcrit ρcrit

the density parameters for all types of ﬂuids, actual and effective. As we stated initially, observational data (such as [15, 259, 412]) have concluded that

Ωrad + Ωb ' 0.04 , Ωcdm ' 0.27 , Ωk ' 0 and ΩDE = ΩΛ ' 0.69 ,

and wDE = −1, which states that approximately 68 % of the Universe’s content in mass or energy is attribute to dark energy, or the Cosmological constant. Con- sequently, the dark energy ideed corresponds to an actual of effective ﬂuid with pressure P = −ρ. 1.3 accelerating expansion and cosmological constant 29

Such fluids are considered to exist within the context of quantum theories, such as the hypothetical fluid corresponding to the energy of vacuum level. This energy is equivalent to the zero-point energy of the Universe -the difference between the classical and the quantum lowest possible energy of the Universe- and is equal to the energy of virtual particles and anti-particles being created and destroyed instantly. According to quantum physics, this vacuum energy behaves like a fluid with negative pressure and barotropic index w = −1. As a result, the quantum vacuum energy could be a candidate for this “dark energy” fluid. In order to explore this possibility, we can simply calculate the values of the Cosmological constant corresponding to the quantum vacuum energy and compare it to the one arising from the observations. The energy density of vacuum is −27 3 −29 3 known to be ρvacuum = 5.96 10 kg/m , or about 10 g/cm ; consequently, the Cosmological constant should be proportional to it, in other words, of the order 2 of MPlanck, so 54 2 Λvacuum = 6 10 eV .

Knowing from observations that ΩΛ = 0.6889 ± 0.0056 and that the current value −18 −1 of the Hubble rate is H0 = 67.66 ± 0.42 (km/s)/Mpc = 2.19277 10 s , we can easily calculate the value of the actual value of the Cosmological constant as

Λ = 1, 1056 10−52 m−2 = 1.30927 10−53 eV2 .

It is obvious that the two numbers do not coincide; in the same system of units, the two Cosmological constant have a discrepancy as high as 120 orders of magnitude, deeming this explanation as unﬁt. The quantum vacuum energy, although it seems to exist and permeate the whole Universe in the same manner as the dark energy ﬂuid, appears with so large a value that, on the one hand it does not seem to contribute to the cosmological equations, and on the other it cannot offer a viable explanation for the observed value of the Cosmological constant that corresponds to this dark energy.

1.3.2 Phantom Dark Energy

Another hypothetical form of dark energy is the phantom energy, that possesses not only negative pressure but also negative kinetic energy, so that the effective equation of state would yield w < −1 (see refs. [18, 142, 203]). Such a behaviour is expected to arise either from some exotic form of matter -a similar hypothesis to the “dark matter” case-, maybe a particle that is yet to be discovered, or from some arbitrary quantum ﬁelds; purely geometric theories have also been employed to provide such a behaviour, close enough to the canonical dark energy, but with we f f < −1. This case is often questioned since it produces ﬁnite-time future singularities, such as the “Big Rip”, whereas the Universe is bound to end die to its rapid 3 expansion tearing apart structures of matter; given, for example, w = − and e f f 3 30 introduction

H0 ' 70 (km/s)/Mpc, we can calculate that in 22 billions of years the Universe will end [80]. Though this case can occur in both General Relativity and modiﬁed theories, it is not considered a valid hypotheses for the future of the Universe [142, 373], hence the phantom dark energy is dismissed. Many evidence from observations can also be accounted for in order to dispose such theoretical ideas, although their viability cannot be completely excluded [67, 257].

1.3.3 Quintessence

Finally, the Quintessence theories focus on the fact that the accelerating expansion occurs late in the evolution of the Universe -as if the mechanism causing it was absent in all previous stages- and behaves as a constant “force”, easily parametrised via a constant, Λ. Generally, it can be realistic for the Universe to contain this mechanism from its initiation, but show it only in the late-time stage, since other components dominated the previous cosmic eras and deterred its emergence; subsequently, no actual problem arises as for the appearence of the “dark energy” only in the latest era, so long as typical baryon matter, cold dark matter and relativistic matter (radiation) can dominate it in all previous eras. This, however, was not enough for some cases, since a time-varying “dark energy” sounded equally good (see refs. [81, 82, 336] as first attempts to this direction). Eventually, a scalar field, Q or Quintessence, can be perceived as causing the late-time accelerating expansion. This scalar field acts under the potential V(Q) and has effective energy density 1 ρ = Q˙ 2 + V(Q) , Q 2 and effective pressure 1 P = Q˙ 2 − V(Q) ; Q 2 so, the effective equation of state corresponding to this scalar field is

1 ˙ 2 P Q − V(Q) w = Q = 2 ,(48) Q ρ 1 Q Q˙ 2 + V(Q) 2

which generally evolves in time, as does the scalar field. The variation of wQ over time allows for values different from wDE = −1, that correspond to the canonical dark energy, or to the Cosmological constant scenario. A favourable perception is that quintessence has an energy density that closely tracks the radiation mass- energy density up to the matter-radiation equality moment, whereas the “dark energy” characteristics are triggered and quintessence gradually grows to wQ ∼ −1 in the late-time, where it dominates matter and thus the whole Universe. Again, many scalar fields could nominate for the role of quintessence; geometric terms or holographic terms could also do it. Aside from the holographic dark 1.4 past and future finite-time singularities 31 energy, that arises from the quantum fluctuations of space-time, a very famous model is that of the kinetic quintessence (k-essence), that can arise from Lovelock or Horndeski theories of gravity; another proposed model, the Quintom scenario, attempts to combine the quintessence with phantom fields features and separate the cosmological models on the basis of the w = −1 boundary for the late-time acceleration.

1.4 past and future finite-time singularities

Let us assume that the Hubble rate contains a singularity at some ﬁnite-time ts. By this we mean that all trajectories (geodesics) are incomplete as they approach this time; the evolution of some phenomena according to an observer at rest, is not capable to continue. This can be easily parametrised if we consider a Hubble rate of the form h = 0 H β ,(49) (ts − t) where h0 and β are real constants. The scalar curvature with respect to time is

 12h2  0 whenβ > 1  (t − t)2β  s 12h2 − 6h R = 6H˙ + 12H2 = 0 0 whenβ = 1 .(50) ( − )2  ts t  6h − 0 <  β+1 whenβ 1 (ts − t)

The above curvature, for different values of β corresponds to different behaviour of the 3-d space, and thus it leads to different types of singularities. Examining carefully, we see the following four cases (see see Cotsakis and Klaoudatou (2004) for a rigorous discussion ref. [289] for a complete list of refer- ences):

1. Type I or “Big Rip” singularity: this type stands for β ≥ 1. The singularity occurs at t → ts, as the scale factor tends to inﬁnity (a → ∞) along with the effective energy density and pressure (ρ → ∞ and |P| → ∞). It is among the most usual and well-studied case (see for example ref. [309]). (Chimento and Lazkoz, 2004; Briscese et al., 2007; Bouhmadi-López et al., 2015).

2. Type II or “Sudden Future” singularity: this type corresponds to 1 > β > 0. The singularity occurs at t → ts, for a bounded scale factor (a → as), as the effective energy density reaches a constant value (ρ → ρs) and the pressure increases to inﬁnity (|P| → ∞). (Barrow, 2004a; Barrow and Tsagas, 2005; Barrow and Cotsakis, 2013).

3. Type III or “Big Freeze” singularity: this type complies with 0 > β > −1. The singularity occurs at t → ts, for a bounded scale factor (a → as), as 32 introduction

the effective energy density and pressure increase to infinity (ρ → ∞ and |P| → ∞). It is implied that the a bounded expansion of the Universe is compatible with an infinite increase in the mass and energy of all matter fields and modified gravity terms, so the latter must somehow drive the former. (Bouhmadi-López et al., 2007; Yurov et al., 2008; Ashtashenok et al., 2012).

4. Type IV singularity: this type corresponds to β < −1 (proposing that it is not an integer). It occurs at t → ts, for a bounded scale factor (a → as), as the effective energy density and pressure tend to zero (ρ → 0 and |P| → 0). It is implied that the a bounded expansion of the Universe is compatible with a vanishing of all mass and energy of all matter ﬁelds and modiﬁed gravity terms, so the latter must somehow balance out the former. (Barrow, 2004b; Barrow and Tsagas 2005).

Here ρe f f and Pe f f are deﬁned as in eq. (111). 1.5 a notice on notions and notation 33

1.5 a notice on notions and notation

Before we proceed, it is useful to make a short comment on the notions and notation used so far, that will be used in the dissertation as follows. The main geometric features employed in relativistic gravity and cosmology, hence in the modified theories of gravity and cosmology as well, are the tensors. Tensors are geometric objects that map other objects to itself in a multi-linear manner; given a metric affine manifold with a coordinate basis, tensors can be expressed as multidimensional arrays, whose elements correspond to a mapping on the specific basis. Such an object is expressed with indices in the form

α1α2...αn T κ1κ2...κm , where n + m is the rank of the tensor. The simplest form of tensors are the scalars (0-rank) and the vectors (1-rank). The full definition of tensors is given by means of a coordinate basis change, since these objects remain unaffected, or rather invariant in such changes, proposing that the manifold on which they are defined in affine. As a result, given an “old” coordinate basis, {xµ}, and a “new” one, {x˜µ}, along with the (reversible) transformation rules, x˜µ = x˜µ (xν), any tensor follows the following transformation rule, α α λ λ ∂x˜ 1 ∂x˜ 2 ∂x 1 ∂x 1 β β ... ˜ α1α2... 1 2 T κ κ ... = ...... T . 1 2 ∂xβ1 ∂xβ2 ∂x˜κ1 ∂x˜κ1 λ1λ2... Any other multidimensional array of arithmetics that does not follow this transformation rule during a change of the coordinate basis, is not considered an invariant of the manifold and is not a tensor. The indices of the tensors can be upper or lower, depending on whether they correspond to the tangent or the cotangent space defined by the coordinate basis on the manifold. Upper indices correspond to the tangent space, that is defined by means of the coordinate curves tangent on the unit vectors; lower incides correspond to the cotangent space, that is defined by means of the coordinate surfaces vertical to the unit vectors. Greek letters will be used for the indices of tensors that are defined on 4-d pseudo-Riemannian or Einstein manifolds (that have non-degenerate metric and curvature proportional to it), used in General Relativity, where α = 0 denotes the temporal components and α = 1, 2, 3 denote the spatial components; latin letters will be used for the indices of tensors defined on 3-d Riemannian manifolds (that have positively defined metric and curvature), where i = 1, 2, 3 correspond to spatial components only. Since the analysis is conducted on curved differentiable manifolds, many forms of differentiation shall appear. Partial derivative with respect to the coordinate basis shall be denoted as ∂U = ∂ U . ∂xµ µ Thus, the covariant derivative is defined as the derivative of a tensor along the tangent curves of the manifold; it is an extension of the partial derivative, equal 34 introduction

to it when scalars are considered and diverging from it when higher-order tensors are differentiated. The divergence results from the affine connection of the manifold, measuring the latter’s divergence from a flat Euclidean space, and is as large as the order of the differentiated tensor. More specifically, given a scalar, Φ, we have ∇µΦ = ∂µΦ ; µ given a vector, V in the tangent space and Vµ in the cotangent space, we have

ν ν ν λ λ ∇µV = ∂µV + Γ µλV and ∇µVν = ∂µVν − Γ µνVλ ;

µν µ ﬁnally, given a 2-rank tensor, T , or Tµν, or T ν , we have

ρσ ρσ ρ λσ σ ρλ ∇µT = ∂µT + Γ µλT + Γ µλT , λ λ ∇µTρσ = ∂µTρσ − Γ µρTλσ − Γ µσTρλ and ρ ρ ρ λ λ ρ ∇µT σ = ∂µT σ + Γ µλT σ − Γ µσT λ ,

α and so on. Γ βγ is the affine connection of the manifold, a non-tensor object (see next chapter). Defining a vector along with null spatial components, as the velocity vector of the rest-frame (the velocity vector of the motionless observer), uµ, we may obtain derivatives along the time dimension, that are simply derivatives with respect to time. Such derivatives will be denoted with dots over the specific symbols, e.g.

dU U˙ = = uµ∇ U . dt µ The remaining components of the covariant derivative, that are projected verti- cally to the velocity onto the spatial subspace, correspond to derivatives with respect to spatial coordinates, or

∂U D U = = (gµν − uµuν) ∇ U , µ ∂xµ ν where gµν the metric of the manifold -this equation is valid only for a scalar quantity, U, and in general the projection tensor hαβ = gαβ − uαuβ must be multiplied k + 1 times to the covariant derivative, where k is the rank of the tensor ﬁeld U. If, however, the differentiation is made with respect to some other notion of time, e.g. the e-foldings number or the conformal time, then it will be denotes with a tone, as dU dt U0 = = U˙ , dN dN using the chain rule of differentiation. Part II

AROADMAPTOMODIFIEDTHEORIESOFGRACITY

The following modiﬁed theories are brieﬂy presented, so as to achieve a complete introduction to the respective literature. 1. The f (R) theory of gravity 2. The f (G) and the f (R, G) theories of gravity

THE f (R) THEORYOFGRAVITY 2

To trace the roots of the f (R) theory of gravity, we should go back to Has Adolph Buchdahl and 1970 [73], when a form of the theory was first proposed, and to Alexei Starobinsky almost a decade later [379, 380], who constructed a viable inflationary model based on this extension of General Relativity. The key idea of the theory is to maintain a more abstract form in the Lagrangian so as to be more flexible describing the specific effects of gravity in small and large scales. One of the key issues noted quite early is that quantum corrections in General Relativity would produce higher-order terms of scalar curvature in the action (such as the R2 in the Starobinsky model); furthermore, such terms could be used in order to describe additional features of the original theory, such as the Cosmological R2 constant (equal to − in the Starobinsky model). 3M2 Consequently, the f (R) theory has been usually employed to deal with the early- [47, 193, 234, 281, 291, 314, 363, 390] or late-time accelerations [232, 282, 332, 352], or the reheating after inflation [278]; unifications of both stages has also been worked out (see refs. [29, 112, 296, 302, 324, 338, 351] inter alia). This strand of modified theories of gravity may be considered both the most re- silient to observational data, but also the most popular among the many in different variants in the literature (see, for example refs. [47, 314, 348]. Its formulation and key aspects as an alternative to Λ-CDM cosmology have been discussed in numerous reviews, such as [46, 86, 144, 289, 295, 303] and [119, 374–376, 378], and based on these we shall try to demonstrate the fundamental features of the theory and the main conclusions derived from it concerning the validity and viability of the respective cosmological models.

2.1 general properties

2.1.1 The metric tensor

Defining an affine metric manifold, where a basis of vector fields {vα} or frame is mapped on the coordinate system {xα}, such that

∂ v = {vα} = (51) ∂xα the metric tensor is deﬁned as ∂ ∂ gµν = g (vµ, vν) = g , ,(52) ∂xµ ∂xν

37 38 the f (r) theory of gravity

where g(v, w) is a one-form mapping from two vector ﬁelds, v and w, tangent on a speciﬁc point of the manifold. Proposing that the space has a coordinate system, then

µν −1 µ ν gµν = g eµ, eν = eµeν , g = g eµ, eν = e e and µ µ µ (53) g ν = I eµ, eν = e eν = δ ν ,

α µ where {e } the unit vectors of the basis and δ ν the Krönecker tensor (for which µ µ δ µ = 1 and δ ν = 0 if µ 6= ν). In other words, the components of the metric tensor represent the scalar products of the components of the unit vectors. By means of the metric, the vectors deﬁned on a manifold can be transformed from their contravariant to their covariant form and backwards, hence from their representation with respect to the tangent vector basis to their representation with respect to the dual cotangent vector basis, or reciprocal basis. In simpler terms, if {vα} the covariant components of a vector, represented in terms of the tangent basis {eα} along the coordinate curves of the space (manifold), and {vα} the contravariant components of it, represented in terms of the dual cotangent basis {eα} along the coordinate surfaces of the space (manifold), then the transformation between the two is expressed as

β α −1 α αβ vα = g (eα, v) = gαβv and v = g (e , v) = g vβ .(54)

In the same manner, the inner product of two vectors, {vα} and {wα}, can be expressed with the intervention of the metric tensor as

µ ν µ ν µν vw = gµνv w = v wµ = vνw = g vµwν .(55)

The same two uses of the metric tensor can be easily generalised for any tensor ﬁeld of (k, l)-rank, e.g.

κ ν κν λν µν λν µ Tµν = gκµT ν , Tµ = gµκ T = g Tµλ andT = g T λ .(56)

The metric tensor satisﬁes by deﬁnition the following three conditions:

1. The metric is bilear: Given three tangent vector ﬁelds, {yα} , {uα} and {vα} at a point p, and two scalars, a and b, then

g (yµ, auν + bvν) = ag (yµ, uν) + bg (yµ, vν) and g (auµ + bvµ, yν) = ag (uµ, yν) + bg (vµ, yν) .

In a coordinate system, the above could be expressed as the bilinearity of the vector ﬁelds deﬁned tangnet (or cotangent) on the manifold, expressed in the bilinear identity of their arithmetic operations,

µ µ µ µ ν µν y (auµ + bvµ) = ay uµ + by vµ = agµνy u + bg yνvµ .(57) 2.1 general properties 39

2. The metric is symmetric: Given two tangent vector ﬁelds {vα} and {wα}, then g (vµ, wν) = g (vν, wµ) . The above is summarised in a coordinate space as the inverse of the indices,

µν νµ g = g and gµν = gνµ (58)

3. The metric is non-degenerate: Given two tangent vector ﬁelds {vα} and {wα} and considering that one of them is non-zero (vα 6= 0), then for every wα a speciﬁc g (vµ, wν) always exists. In a coordinate system, if vα 6= 0 and vw = 0, then

αβ α g wβ = 0 −→ w = 0 . (59)

From this, we deduce that the determinant of the metric tensor is always non-zero, g = det(gµν) 6= 0 . (60)

The metric is used to deﬁne the line element of the manifold

2 µ ν ds = gµνdx dx ,(61) where {dxα} deﬁnes unit vectors {eα} in a coordinate basis and ds the proper distance on the manifold. Furthermore, it is used to deﬁne the (super-)volume element s n dV = dnx = |g| ∏ dxn ,(62) i=1 where n the dimension of the manifold.

2.1.2 The Afﬁne Connection: torsion-free Levi-Civita case

The manifold described so far is metric and can be restricted even more, with the choice of torsion-free afﬁne connection. In this case, considering the unit vectors {eα} of the basis, the connection of the manifold my be deﬁned as

λ ∇µeν = Γ µνeλ ,(63) where ∇α is the covariant derivative in the coordinate direction of eα but deﬁned on the general manifold, whereas eα = ∂α is a local (holonomic) basis. Demanding that the metric tensor is covariantly constant, e.g. that its covariant derivative is identically zero, we obtain

λ λ ∇κ gµν = ∂κ gµν − gµλΓ νκ − gνλΓ µκ = 0 , 40 the f (r) theory of gravity

from where the Levi-Civita connection is given in closed form with respect to the metric tensor, 1 Γκ = gκλ ∂ g + ∂ g − ∂ g .(64) µν 2 µ λν ν λµ λ µν The Levi-Civita is a non-tensor quantity, also known as the Christoffel symbols. The Levi-Civita connection satisﬁes speciﬁc properties.

1. The Levi-Civita connection, being torsion-free, is symmetric in the lower two (second and third) indices,

κ κ Γ µν = Γ νµ .(65)

This is the only possible case, where all elements of the connection may vanish at a point of the manifold, following a speciﬁc coordinate system. If -for any reason- the connection is non-symmetric, then it remains non- symmetric under any change of the coordinates; hence, no point exists in the manifold, where the connection will be zero.

2. Generallly, the Levi-Civita connection is not a tensor, as it is clearly demonstrated in the transformation from a coordinate system, {xα}, to another, {x¯α}, ∂x¯κ ∂xρ ∂xσ ∂2xλ ∂x¯κ Γ¯ κ = Γλ + , µν ∂xλ ∂x¯µ ∂x¯ν ρσ ∂x¯ρ∂x¯σ ∂xλ where the second term of the right-hand side is inhomogeneous and would not exist if the connection was a tensor. It is proved, though, that if the transformation from one coordinate system to another is linear, then the inhomogeneous part of the transformation vanishes identically; in this case, the Levi-Civita connection behaves like a tensor, though it is not really one. α ˜ α 3. Given two ﬁelds of connections, Γ βγ and Γ βγ, then their difference is a tensor since it transforms normally (the inhomogeneous parts of the two isolated transformations are equal and hence they cancel out each other). Here, we should notice that the inhomogeneous term of the transformation depends solely on the change between coordinate systems and not on the Christoffel symbols themselves; arguably, the non-tensor behaviour of the connection should be attributed to the coordinate basis of the manifold.

The Christoffel symbols may also be contracted in the following two ways; initially, we may contract any of the two lower indices with the ﬁrst one, p 1 ∂g ∂ log |g| Γµ = Γµ = = ;(66) µν νµ 2g ∂xν ∂xν

afterwards, we may contract along the second and third indices,

p κλ µν κ 1 ∂ |g|g g Γ µν = − p .(67) |g| ∂xλ 2.1 general properties 41

Finally, one should notice that, by means of the affine connection, we may define the autoparallel and geodesic (zero-acceleration) trajectories on the manifold. Given a vector vµ of the coordinate basis, that is tangent on the coordinate curves and may represent the velocity of a body moving along these curves, its total derivative with respect to some parameter τ (the proper distance or the proper time) is by definition set equal to zero, Dvµ dxν = ∇ vµ = 0 , Dτ dτ ν and, using the definition of the covariant derivative, we obtain dvµ + Γµ vκvλ .(68) dτ κλ Since, the vector vµ is considered the velocity of a body moving along the coor- dxµ dinate curves (with zero acceleration), then vµ = , and hence the geodesic dτ trajectories are defined as

d2xµ dxκ dxλ + Γµ .(69) dτ2 κλ dτ dτ In the case of an afﬁne metric (torsion-free) manifold, the geodesic trajectories coincide with the autoparallel ones; this is not the case when torsion or non- metricity appears. In the physical world, as described by Einstein’s General Rel- ativity, the geodesic curves correspond to zero-acceleration curves, and hence to the trajectories of free-falling bodies in the curved space-time; the autoparallel curves correspond to the parallel transport of a vector (e.g. the velocity of a body) that preserves the length of it.

2.1.3 Curvature

The curvature of a manifold is associated with the Riemann-Christoffel curvature tensor, that is deﬁned as a linear transformation of the tangent space of the manifold, associated with the transport of tangent vectors along the manifold while always parallel to other tangent vectors. Given a coordinate basis, {xα}, the trans- lation of a vector, {wα} along the coordinate curves deﬁnes the curvature tensor

α R κλµwα = ∇κ∇λwµ − ∇λ∇κwµ ;(70) the latter is also known as the Ricci identity. Thus, the curvature tensor measures the extent to which the metric tensor is not locally isometric to that of a ﬂat Euclidean space.

2.1.3.1 The Riemann-Christoffel tensor In any afﬁne metric manifold, the Riemann-Christoffel tensor represents the totality of curvature, measured as stated above. The deﬁnition by means of the Bianchi 42 the f (r) theory of gravity

identities may eventually lead to the following simpler deﬁnition, by means of the Levi-Civita connection.

α α α α β α β R κλµ = ∂λΓ κµ − ∂κΓ λµ + Γ λβΓ κµ − Γ κβΓ λµ .(71) Utilising the interlinks of a Christoffel symbol to the metric tensor and lowering the ﬁrst index of the Riemann-Christoffel tensor, we may ﬁnally reach a clear association of the curvature tensor to the metric tensor 1 R = − ∂ ∂ g + ∂ ∂ g − ∂ ∂ g − ∂ ∂ g .(72) αβµν 2 ν β αµ µ α βν µ β αν ν α βµ From the above formulations of the Riemann-Christoffel tensor, the following symmetries and identities arise:

1. Skew symmetry: the curvature tensor is antisymmetric in the ﬁrst two and the last two indices respectively,

Rαβµν = −Rβαµν = −Rαβνµ = Rβανµ .(73)

2. Interchange symmetry: the curvature tensor is symmetric in the interchange of the ﬁrst two indices with the last two,

Rαβµν = Rµναβ (74)

3. Cyclic Permutation Sum or Algebraic (ﬁrst) Bianchi Identity: following the combination of the above symmetries, the curvature tensor is antisymmetric in all last three indices, in other words it follows the cyclic permutation sum identity, 1 R = R + R + R = 0 . (75) α[βµν] 6 αβµν αµνβ ανβµ 4. Differential (second) Bianchi Identity: in the same manner, the covariant derivatives of the curvature tensor are also antisymmetric in the last three indices, 1 ∇ R = ∇ R + ∇ R + ∇ R = 0 . (76) [λ| αβ|µν] 6 λ αβµν ν αβλµ µ αβνλ

The Riemann-Christoffel tensor can be broken down to simplest components, each of whom represents a different aspect of curvature, in the short- or the long- αβ µν range. Naming Rµν = g Rαµβν the Ricci tensor, R = g Rµν the Ricci scalar and Cαβµν the Weyl tensor, the Riemann-Christoffel tensor is written as 1 R = g R − g R − g R + g R − αβµν n − 2 αµ βν αν βµ βµ αν βν αµ (77) 1 − Rg g − g g + C , (n − 1)(n − 2) αµ βν αν βµ αβµν where n the dimension of the manifold. 2.1 general properties 43

2.1.3.2 The Ricci tensor and the Ricci scalar As stated, the Ricci tensor is deﬁned as the non-zero trace of the Riemann tensor,

µν µ Rαβ = g Rµανβ = R αµβ ;(78) it is noted that of all possible contractions of the Riemann-Christoffel tensor, this is the only one that survives in the case of an afﬁne metric (and torsion-free) manifold. From the deﬁnition of the Riemann-Christoffel tensor with respect to the Levi-Civita connection (eq. (71)), we may express the Ricci tensor in terms of the connection as

µ µ µ ν µ ν Rαβ = ∂µΓ αβ − ∂βΓ αβ + Γ αβΓ µν − Γ ανΓ βµ ,(79) while utilising the the expression of the Riemann-Christoffel tensor with respect to the metric tensor (eq. (72)), we may also write the Ricci tensor in term of the metric as 1 R = − ∂µ∂ g .(80) αβ 2 µ αβ From the above, it is easy to see that the Ricci tensor is symmetric

Rαβ = Rβα , which is derived both from the symmetries of the Riemann-Christoffel tensor, but also from the algebraic Bianchi identity. As for the differential Bianchi identity, it can be contracted to

ν ∇µRαβ − ∇βRαµ + ∇νR αβµ = 0 . (81)

The trace of the Ricci tensor,

αβ αβ µν R = g Rαβ = g g Rαµβν ,(82) has been identified as the Ricci scalar, or as the scalar curvature. It should be stated that, both the Ricci tensor and the Ricci scalar represent a specific perception of the total curvature of the manifold; we could name it the “local” curvature, or the short-range divergence from a Euclidean space. In this, we mean that any information inherited from the Riemann-Christoffel tensor to the Ricci tensor and hence to the Ricci scalar is associated with the effects of curvature locally around the point of the manifold, where the afore-mentioned tensors were defined. Think- ing of a body that moves along a geodesic of the manifold, the Ricci curvature measures the effect on the change of the body volume. From the differential Bianchi identity (eq. (76)), we may obtain the “gradient” of the scalar curvature in the form of

ν ∇µR = 2∇νR µ .(83) 44 the f (r) theory of gravity

From this point, we may define a “gradient”-free tensor that will encapsulate the essential of “local” curvature, as 1 G = R − Rg .(84) µν µν 2 µν This tensor is known as the Einstein tensor and the fact that it is covariantly constant, µν ∇µG = 0 , is a generalised form of a Laplace differential equation for the curvature of a space-time, and hence for gravity. Establishing the equivalence between curvature and gravity in Einstein’s sense, this identity automatically leads to the conservation of energy and momentum, and thus it is indeed a generalised form of the Laplace equation for the gravitational field. Specifically, considering a tensor containing all information about matter fields, their stress, energy and momentum, Tµν, the Einstein tensor is proved to be proportional to it,

Gµν = kTµν ,(85)

8πG where k = is Einstein’s gravitational constant, G is Newton’s gravitational c4 constant and c the speed of light -both of the latter considered constant within the framework of General Relativity. Given the Bianchi identity, we obtain that the energy-momentum tensor is covariantly constant,

µν ∇µT = 0 . (86)

The latter implies that energy and momentum are conserved in the curved manifold of the General Theory of Relativity.

2.1.3.3 The Weyl tensor Understanding the Ricci tensor as the trace of the Riemann-Christoffel tensor, and consequently as the components associated with the “local” effects of curvature in the manifold, we should address the Weyl tensor as the trace-free (diagonal) elements of the Riemann-Christoffel tensor, encapsulating the information associated with the long-range effects of curvature. As an example, the Weyl tensor measures the distortion due to tidal forces that a body moving along a geodesic suffers. The deﬁnition of the Weyl tensor is non-else than the abstraction of the traces (the Ricci curvature) from the Riemann-Christoffel tensor, 1 C =R − g R − g R − g R + g R + αβµν αβµν n − 2 αµ βν αν βµ βµ αν βν αµ (87) 1 + Rg g − g g . (n − 1)(n − 2) αµ βν αν βµ 2.1 general properties 45

It is noted that the Weyl tensor is generally zero in manifolds of dimension less than 3 (n < 3). Inherited from the Riemann-Christoffel tensor, the main symmetry of the Weyl tensor lies with the antisymmetry of the ﬁrst two and the last two indices respectively, Cαβµν = −Cβαµν = −Cαβνµ = Cβανµ . From its deﬁnition, the Weyl tensor is traceless, and thus

α α C αβγ = C βαγ = 0 .

Finally, the Weyl tensor preserves the Algebraic Bianchi identity and also presents a modiﬁed Differential Bianchi identity; the ﬁrst one is merely the cyclical permutation sum with respect to the last three indices,

1 C = C + C + C = 0 . (88) α[βµν] 6 αβµν αµνβ ανβµ As for the second Bianchi identity, we can express it as

2(n − 3) 1 ∇ Cα = ∇ R − Rg = α βµν n − 2 [µ ν]β 2(n − 1) ν]β (89) n − 3 1 = ∇ R − ∇ R − g ∇ R − g ∇ R . n − 2 µ νβ ν µβ 2(n − 1) νβ µ µβ ν

Noting the equivalence between curvature and gravity, and thus the proportion- ality of the Ricci curvature to the energy-momentum tensor, as given in General Relativity by the ﬁeld equations (eq. (85), we may obtain the following propagation equation for the Weyl curvature,

n − 3 1 ∇ Cα = κ ∇ T − ∇ T − g ∇ T − g ∇ T .(90) α βµν n − 2 µ νβ ν µβ (n − 1) νβ µ µβ ν

Much like the electromagnetic field tensor (the Faraday tensor) that decom- poses to an electric field vector and a magnetic field vector, the Weyl tensor can be decomposed to an electric field tensor and a magnetic field tensor. These are µ ν µ ν defined in manifolds where Rαµβνu u 6= 0 and subsequently Cαµβνu u 6= 0 fro any tangent vector vα; their definitions arise exactly from this ansatz concerning the Weyl tensor and its Hodge dual. Specifically, the electric field tensor is given as µ ν Eαβ = Cαµβνu u ,(91) where uα the velocity of a body moving along a geodesic; in this manner, the gravito-electric field measures the non-convergence of geodesics, and thus the evolution of bodies on such non-converging trajectories. The magnetic field, on the other, is defined as 1 H = ε C κλuµuν ,(92) αβ 2 βνκλ αµ 46 the f (r) theory of gravity

where εκλµν is the Levi-Civita antisymmetric symbol. The Weyl tensor can be expressed in terms of those two, as

µν = [µ ν] + [µ ν] − κ λ[µ ν] − µνκλ Cαβ 2u[αEβ] u δ[α Eβ] εαβκλu H u ε uκ Hλ[αuβ] .(93)

The electric and magnetic tensors of curvature are symmetric,

Eαβ = Eβα and Hαβ = Hβα ,

and trace-free,

αβ α αβ α g Eαβ = Eα = 0 and g Hαβ = Hα = 0 .

Furthermore, in a pseudo-Riemannian space-time, their components are spatial and hence vertical to the velocity vector,

β β Eαβu = 0 and Hαβu = 0 .

2.2 field equations and conservation laws

The standard f (R) theory begins from an action of the form

Z p 1 S = d4x −g f (R) + L ,(94) f (R) 2κ m f (R) L = where√ is an arbitrary and unknown function of the Ricci scalar and m −gLm is the Lagrangian density of the matter ﬁelds. In order to acquire the equations of motion of this theory, as in the General Relativity, we need to variate the action with respect to the metric, gµν. A usual consideration of the function f (R) is as a series of terms around f0(R) = R or f1(R) = R − 2Λ, that result to the General Relativity, a a f (R) = ... 2 + 1 + (R − 2Λ) + b R + b R2 + ... , (95) R2 R 1 2

where the ai and bi parameters can be obtained by compering with empirical results [378]. The variation of the determinant of the metric is easily obtained as

p 1p δ −g = − −gg δgµν .(96) 2 µν µν Having deﬁned the Ricci scalar as R = g Rµν, we may obtain its variation with respect to the metric as µν µν δR = Rµνδg + g δRµν , where λ λ λ λ δRµν = δ ∇λΓ µν − ∇νΓ λµ = ∇λδΓ µν − ∇νδΓ λµ 2.2 field equations and conservation laws 47 and 1 1 δΓλ = gκλδ ∇ g + ∇ g − ∇ g = gκλ ∇ δg + ∇ g − ∇ δg ; µν 2 µ νκ ν µκ κ µν 2 µ νκ ν µκ κ µν hence, the variation of the Ricci scalar is

µν αβ µν µν δR = Rµνδg + gµνg ∇α∇βδg − ∇µ∇νδg .(97)

Eventually, the variation of the action (eq. (94)) with respect to the metric, yields

Z 1 Z δS = d4x p−gδ f (R) + f (R)δp−g + d4xδL = 2κ m Z 1 Z = d4x p−gF(R)δR − p−gd f (R)δgµν + d4xδL = 2 µν m Z 4 p µν αβ µν µν = d x −g F(R) Rµνδg + gµνg ∇α∇βδg − ∇µ∇νδg − 1 Z − f (R)g δgµν + d4xδL , 2 µν m ∂ f (R) ∂L where F(R) = . Acknowledging that T = m is the energy-momentum ∂R µν ∂gµν tensor and integrating by parts on the second and third terms of the ﬁrst integral, we obtain Z p 1 δS = d4x −gδgµν F(R)R + g gαβ∇ ∇ F(R) − ∇ ∇ F(R) − f (R)g δgµν + T δgµν . µν µν α β µ ν 2 µν µν Demanding that the action is invariant with respect to the metric, so its variance δS with respect to it is zero, = 0, we obtain the following equations of motion, δgµν essentially the ﬁeld equations of the f (R) theory,

1 h i F(R)R − f (R)g + g gαβ∇ ∇ − ∇ ∇ F(R) = κT .(98) µν 2 µν µν α β µ ν µν As with Einstein’s field equations, the left-hand side of eq. (98) contains the geometric terms and thus it denotes the curvature of the space-time associated with the presence of matter fields, while the right-hand side contains the energy- momentum tensor, standing for the total mass, energy and momentum of the matter fields that cause the afore-mentioned curvature (and exist because of it). The above can be rewritten as 1 h i F(R)G + (RF(R) − f (R)) g + g gαβ∇ ∇ − ∇ ∇ F(R) = κT ,(99) µν 2 µν µν α β µ ν µν where Gµν is the Einstein tensor; the appearance of the latter in the field equations allows us to separate even further the classical relativistic effects of curvature -as depicted on the Einstein tensor- from the modifications imposed by the f (R) theory -depicted in the remaining terms of the left-hand side. 48 the f (r) theory of gravity

According to ref. [337] the Nöther symmetries, responsible for the conservation laws of a physical theory, are consistent with modified gravity theories, hence the laws of conservation of energy and momentum apply still in the f (R) theories. Specifically, [88] provided us with a general formula that secures that these conservation laws are valid in the context of any modified theory of gravity; according to him, any modified theory of gravity whose equations of motion can be written in the form of eq. (99), where the relativistic curvature (Einstein tensor) and the matter fields (energy-momentum tensor) are separated from the additional fields or geometric features, have the exact conservation laws that General Relativity has. µν ∇µT = 0 . (100) For this to hold, specific “energy” conditions must also hold, concerning the remaining terms appearing in the left-hand side of the field equations. Specifically, in the case of the f (R) theory, the “energy” conditions imply that

∇βF(R) 1 (RF(R) − f (R)) gαβ − gαµgβν∇ ∇ F(R) + gαβgκλ∇ ∇ F(R) = F(R) 2 µ ν κ λ 1 = F(R)∇ R + R∇ F(R) − ∇ f (R) gαβ− 2 β β β αµ βν αβ κλ − g g ∇β∇µ∇νF(R) + g g ∇β∇κ∇λF(R) , and after some simplifications, 1 ∇ F(R)Gαβ = (F(R)∇ R + R∇ F(R) − ∇ f (R)) gαβ .(101) α 2 α α α From this, we may derive a constraint for the form of the f (R) function, so that the respective models are viable from the perspective of the conservation of energy and momentum -as we know it; furthermore, we may derive the conditions for which the f (R) gravity acts like the relativistic gravity, namely it preserves the attractive behaviour. Ref. [88] calculates the condition for attractive f (R) gravity in the case of a perfect fluid dominating the Universe; given that ρ = T00 the mass-energy density and P = Tii the pressure of the perfect fluid, the condition writes

µν κT = κ(ρ + 3P) ≥ (RF(R) − f (R)) − h ∇µ∇νF(R) + 3F¨(R) .(102)

It is also noted that in vacuum f (R) gravity may become repulsive since

µν (RF(R) − f (R)) − h ∇µ∇νF(R) + 3F¨(R) ≤ 0 .

Several exact solutions may by found for the f (R) theory, pretty much in the manner of General Relativity. Two examples of symmetric space-times are considered [289, 378], one with notable astrophysical signiﬁcance and the other with special cosmological implications. 2.2 field equations and conservation laws 49

First, we shall assume a vacuum solution, so Tµν = 0, and that the Ricci tensor is covariantly constant, and by result proportional to the metric, Rµν ∝ gµν. As a result of these, the field equations (eq. (98)) are simplified into 2 f (R) − RF(R) = 0 , (103) that satisfy a solution in the form ds2 = −k(r)dt2 + l(r)dr2 + r2dθ2 + r2 sin2 θdφ2 . From this point, it is easy to determine k(r) and l(r) so that the Schwarzschild - (anti-)deSitter solution to be obtained, 2GM r2 2GM r2 −1 ds2 = − 1 − ∓ dt2 + 1 − ∓ dr2 + r2dθ2 + r2 sin2 θdφ2 , r L2 r L2 (104) where M the “mass” of the Schwarzschild black hole and L the “length” parame- 12 ter of the (anti-)deSitter expansion, associated with curvature as R = ± ; notice L2 that the positive sign indicates the deSitter solution, while the negative sign corresponds to the anti-deSitter one. In a similar way, we may derive the Kerr - (anti-)deSitter solution. Second, we may consider a flat Friedmann-Robertson-Walker space-time, where the line element must be given as

3 2 ds2 = −dt2 + a(t)2 ∑ dxi ,(105) i=1 where a(t) is the expansion factor of the 3-d space (the Universe). We also consider an ideal fluid of mass-energy density ρ and pressure P, that fills the space- time, whose energy-momentum tensor is Tµν = ρuµuν + P(uµuν + gµν) ,(106) where uµ the velocity vector; in the inertial frame of reference comoving with the fluid, the energy-momentum tensor is a diagonal matrix, with density being the temporal component and pressure dominating the spatial ones. Defining the a˙ Hubble rate as H = , and acquiring the scalar curvature as R = 12H2 + 6H˙ , a we may introduce the above in the field equations (eq. (98)) and obtain the f (R)- version of the FRW cosmological equations, 1 3 F(R)H2 + F˙(R)H + ( f (R) + F(R)R) = κρ (107) 2 or, expressing everything in terms of curvature, 1 f (R) − 3 H2 + H˙ f (R) + 18 4H2 H˙ + HH¨ f (R) = κρ ,(108) 2 R RR 1 2 2 2 ... f (R) − H˙ + 3H fR(R) + 6 8H H˙ + 4H˙ + 6HH¨ + H fRR(R)+ 2 2 + 36 4HH˙ + H¨ fRRR(R) = −κP ,(109) 50 the f (r) theory of gravity

∂ f ∂2 f ∂3 f where f (R) = F(R) = , f (R) = and f (R) = the derivatives. R ∂R RR ∂R2 RRR ∂R3 We can rewrite the above in a sense as to present the difference between the f (R) theory and General Relativity, by isolating the terms that correspond to explicitly the former from those that correspond to the latter. This is made possible if we consider that f (R) = R + f˜(R) , where f˜(R) encapsulates all differences obtained in the modified theory [289]. From equations (108) and (109), we may reach a new from of equations describing the effective mass-energy density and pressure of the cosmic fluid, that contain both the actual matter-fields terms and the geometric terms diverging from the description of General Relativity,

1 1 ρ =ρ + − f˜(R) + 3 H2 + H˙ f˜ (R) − 18 4H2 H˙ + HH¨ f˜ (R) , e f f matter κ 2 R RR (110)

1 1 2 2 2 ... P =Pmatter + f˜(R) − 3H + H˙ f˜R(R) + 6 8H H˙ + 4H˙ + 6HH¨ + H f˜RR(R)+ e f f κ 2 2 + 36 4HH˙ + H¨ f˜RRR(R) .

This description reveals that the effective mass-energy density and pressure may be still written in the form of the Einstein description as

3 1 ρ = H2 and P = − 3H2 + 3H˙ .(111) e f f κ e f f κ This enables us to write the effective barotropic parameter, essentially the barotropic index of the effective equation of state,

Pe f f 2H˙ = = − − we f f 1 2 .(112) ρe f f 3H

[46] and [289] argue that the ﬂuid representation of the FRW equations is often mistreated and leads the higher-order derivatives of curvature invariants contained in the generalised gravitational ﬂuid are ignored.

2.3 scalar-tensor description and brans-dicke equivalence

Drawing from refs. [86, 289] and [96], we will attempt to show that the f (R) theory of gravity can be written in a scalar-tensor form with the introduction of an auxiliary ﬁeld A and the transformation from the Jordan frame, where all above have been described, to the Einstein frame. Introducing the ﬁeld A, the action of eq. (94) is rewritten as

1 Z p S = d4x −g ( f (A) (R − A) + f (A) + L ) .(113) 2κ A matter 2.3 scalar-tensor description and brans-dicke equivalence 51

It is not difﬁcult to see that variation with respect to the auxiliary ﬁeld A leads to

fAA(R − A) = 0 , therefore A = R, as long as fAA(A) 6= 0; subsequently, the substitution of A = R in the action allows for the reproduction of the original action of eq. (94). In order to transpose from the Jordan frame to the Einstein frame, we need to rescale the metric by a kind of a canonical transformation, such as

σ gµν → e gµν ,(114) where σ = − ln fA(A). In doing so, we obtain the action in the Einstein frame as

1 Z p 3 S = d4x −g R − gµν∂ σ∂ σ − V(σ) + L ,(115) E 2κ 2 µ ν matter where A f (A) ( ) = σ ( −σ) − 2σ ˜ ( −σ) = − V σ e g e e f g e 2 fA(A) fA(A) the potential of the rescaling auxiliary field σ; as a result, the action of the f (R) theory with respect to the Jordan frame is merely the action of a scalar-tensor extension of General Relativity, where the scalar curvature is accompanied by the kinetic term and the potential of a scalar field (σ), that could be the result even of the matter fields. Notice that solving the equation σ = − ln fA(A) yields A = g(e−σ). Due to the canonical scale transformation, the scalar field σ appears to be coupled with usual matter; in that case, the mass of the scalar field can be calculated as 3 d2V(σ) 3 A f (A) 1 2 = = − + mσ 2 4 2 ;(116) 2 dσ 2 fA(A) fA(A) fAA(A) if the mass is not very large, then a large correction appears to the Newton law, as in the case where the accelerating expansion of the universe is attempted and the mass of the scalar field should be that of the Hubble rate, namely mσ ∼ 10−33eV. In order to avoid this problem, the mass can be allowed to vary, so that it will become large enough in regions where the curvature is large or under the presence of matter fluids (e.g. in the Solar system) [191]. Consequently, a screening effect is introduced cusing the force generated by the scalar field to be short-ranged, but in such a way that only massive objects (e.g. the planets) can contribute to the correction; this is nothing more but the familiar Chameleon mechanism [216]. Furthermore, a problem emerging from the scalar-tensor description is the antigravity [331]. Since, the effective gravitational coupling constant is κ κe f f = , fRR(R) 52 the f (r) theory of gravity

the function of f (R) must necessarily be constantly convex, in other words we need to ensure that it has constantly positive second derivative, so that no antigravity regions (or regions where the graviton becomes a ghost) may exist. Finally, we should note that the Jordan frame and the standard f (R) description may end up to completely different results from the Einstein frame and the scalar-tensor desciption. On of such problems is that the accelerating expansion in one frame is not ineed mapped to an accelerating expansion in the other, but it can even lead to a deccelerating one [84]; another one is that the several types of singularities encountered in one frame do not hold the same in the other [39, 69].

The f (R) theory is also proved to be exactly equivalent to another modified theory of gravity, the Brans-Dicke theory [377]. Beginning from action of eq. (113) and understanding that A = R, we may redefine the auxiliary field A as φ = fA(A) and setting V(φ) = A(φ)φ − f A(φ) , the action takes the form 1 Z p S = d4x −gφR − V(φ) + L .(117) met 2κ matter It is easy to see that this is the Jordan frame representation of a Brans-Dicke theory with Brans-Dicke parameter ω0 = 0 and hence no kinetic term. Taking into account the definition of the scalar field, φ, we can transpose to the Einstein frame and conclude to the following action

1 Z p 3 1 Z S = d4x −g φR + gµν∂ ∂ φ − V(φ) + d4xL ,(118) pal 2κ 2φ µ ν 2κ matter

which is the Brans-Dicke theory in the Einstein frame with Brans-Dicke parameter 3 ω = − . The same conclusion can be reached with the utilisation of the Palatini 0 2 formalism, where the connection is considered an independent variable, aside from the metric. It becomes obvious from the variation with respect to the scalar ﬁeld, φ, that results to the equation of motion for it,

dV (2ω + 3)gµν∇ ∇ φ + 2V(φ) − φ = κT ,(119) 0 µ ν dφ

µν µ where T = g Tµν = T µ the trace of the energy-momentum tensor, that the dynamics of the scalar ﬁeld are not trivial (or non-existing), judging from the 3 absence of a kinetic term in the Jordan frame, since ω = − emerges when the 0 2 Einstein frame (scalar-tensor description) or the Palatini f (R) theory are considered (see also ref. [167]). 2.4 viable f (R) theories 53

2.4 viable f (R) theories

The f (R) theory of gravity, as described so far, is proved to offer viable cosmological models that are capable to unify the early- and late-time accelerations of the universe, in essence both the inﬂationary era and the present accelerating expansion, without the intrinsic need of a cosmological constant or some other arbitrary scalar ﬁeld. Such models have been presnted by refs. [293, 296–299] and [2, 70, 374, 375], while their viability has been repeatedly proved theoretically refs. [91, 111, 232, 324, 363] and via observational constraints [41, 115, 267, 326, 341, 352]. According to refs. [289, 303], three important conditions should hold, so that any f (R)-founded cosmological model can be considered viable.

1. In order for an inﬂationary era to be generated, the f˜(R) function must be bounded for large curvatures; the boundary should coincide with the effective cosmological constant, Λi that characterises the early stages of the Universe, lim f˜(R) = −Λi .(120) R→∞

2. In a similar manner, the current value of the f˜(R) function should be very small and asymptotically constant (so with almost zero derivative); in this way the late-time accelerating phase of the Universe is generated.

f˜(R0) = −2R˜ 0 and f˜R(R0) ∼ 0 , (121)

−33 2 where R0 ∼ (10 eV) the current value of curvature and R˜ 0 < R0 a constant. This condition restricts very strictly the value of f˜(R), due to cosmological observations and the presence of matter ﬁelds that should be taken into consideration, but can be relaxed on the side of the derivative, 12 13 f˜R(R), since the time scale is large (10 − 10 years) and thus the derivative needs not to vanish but rather to fulﬁll the actual condition that is −33 4 | f˜R(R0)| (10 eV) .

3. The ﬂat Miknowski space-time should always be a solution to the theory and, furthermore, it must always exist within each separate model, as a special case where curvature tends to zero,

lim f˜(R) = 0 . (122) R→0

The fulﬁllment of all the above conditions is a prerequisite for the construction of a viable f (R) cosmology. As a result, the power-series expansion of f (R) is considered a good approximation of the situation or, even further, the hypergeometric functions as a complete solution; one should also notice some more elegant form, such as the Hu-Sawicki. A typical (phenomenological) consideration aside e from the series-expansion could be a power-law, f (R) ∼ R + f0 + f1R , where f0 54 the f (r) theory of gravity

and f1 are arbitrary constants and R is considered large; here, f0 may vanish, but f1 and a careful choice of e are essential to the viability of the theory, as well as to very speciﬁc issues, such as singularities present in the model. Given the last case, the trace of eq. (98) becomes

µν 3g ∇µ∇ν f˜R(R) = R + 2 f˜(R) − R f˜R(R) − κT ,(123)

which in turn indicates that  µν e−1 R when e < 0 or e = 2 3 f1g ∇µ∇νR = .(124) e (2 − ε) f1R when e > 1 or e 6= 2

As a result, when e < 0 or e = 2, the trace of the ﬁeld equations yields

e R + κ(ρ + 3P) = −2 f0 − f1(R − 3R) ,(125)

and when e > 1 or e 6= 2, it becomes

e R + κ(ρ + 3P) = −2 f0 + f1(1 − e)R .(126)

Any of these form for the trace of the field equations separates the Einstein and matter-fields terms, that are on the left-hand side, from the purely “modified” terms, on the right-hand side, given that f (R) has the afore-mention form; this separation is very convenient when the effect of modified gravities on General Relativity should be measured and compared to the observations. This form for the f (R) will be proved useful, combined with a Hubble rate, in the following discussion about finite-time singularities that occur in viable f (R) cosmologies.

To generate inﬂationary dynamics from a viable f (R) theory and proceed by empirically testing the strength of it, in other words by comparing the theoretical results with observations for the primary stages of the universe, one should encapsulate the theory into some key features, such as the spectral index for primordial curvature perturbations and the tensor-to-scalar ratio, that is the ratio of magnitudes of the tensor to the scalar curvature perturbations; the latter two can be obtained observationally from the cosmic microwave background, and thus the viability of the theory can be further tested. Followin [197], the two observables can be fundamentally constructed from every viable theory of modiﬁed gravity by means of the slow-roll indices and some other measures. Given the simple case of f (R) theory, the slow-roll indices are H˙ H˙ ε = − , ε = 0 , ε = −ε = and 1 H2 2 3 1 H2 (127) ε˙ H˙ H˙ H¨ = − + 1 = − + ε4 3ε1 3 2 2 . Hε1 H H H˙ 2.5 viable f (R, φ, X) theories 55

The observables are found to be

2 nS = 1 − 6ε1 − 2ε3 and r = 48ε1 .(128)

It should be noted that the above, although developed independently by [91, 362] are in fact just a simpliﬁed version of an extended form, that will be presented in the next section.

2.5 viable f (R, φ, X) theories

The a priori combination of the f (R) theory with a scalar field φ and its kinetic 1 term, X = − gαβ∂ φ∂ φ, also presents great interest, concerning the viable cos- 2 α β mological models arising from it, in which the early-time and late-time accelerating stages can be unified. Such models have been considered by [141, 194, 195, 287] and more recently by [174, 222, 281, 317, 320, 417, 418]; the formalisms and main results are (partly) summarised by [86, 289, 295, 376]. The usual goal of em- ploying such a model is to generate inflation as well as the gracefule exit from it by means of the scalar field and its potential and not curvature itself. In what follows, we attempt to present the major differences of the addition of the scalar field, φ, in the f (R) theory.

2.5.1 Canonical scalar ﬁeld description

Given a ﬂat Friedmann-Robertson-Walker metric, as in eq. (105), so that R = 12H2 + 6H˙ , and the following action

Z R S = d4xp−g − L = 2κ φ (129) Z p 12H2 + 6H˙ 1 = d4x −g − gµν∂ φ∂ φ − V(φ) , 2κ 2 µ ν 1 where the kinetic energy X = − gµν∂ φ∂ φ1 minus the potential V(φ) of the 2 µ ν scalar field constitute the total of the Lagrangian density of the matter fields, since matter fields are absent and the scalar field is presented with this form. Therefore, the equations of motion of the theory are simply obtained by variation with respect to the metric (and the gradual break-down to temporal and spatial components of the field equations) and to the scalar field. The energy-momentum tensor corresponding to the scalar field is √ δ −gLφ 1 T = = ∂ φ∂ φ − g gκλ∂ φ∂ φ − V(φ) ,(130) µν δgµν µ ν µν 2 κ λ

1 1 In a ﬂat FRW background, X = φ˙ 2. 2 56 the f (r) theory of gravity

so th energy density and the pressure of the hypothetical ﬂuid representing the scalar ﬁeld are easily obtained as the temporal and the spatial (diagonal) components of the energy-momentum tensor respectively,

1 1 1 ρ = T = φ˙ 2 + V(φ) and P = δijT = φ˙ 2 − V(φ) .(131) φ 00 2 φ 3 ij 2 As a result, it is very easy to obtain the Friedmann-Robertson-Walker equations. Varying the action in eq. (129), we reach to the ﬁeld equations of eq. (98); the temporal component yields the Friedmann equation for the constraint of the Hubble rate, κ κ κ H2 = ρ = φ˙ 2 + V(φ) ;(132) 3 φ 6 3 from the spatial component we obtain the Raychaudhuri equation for the evolution of the Hubble rate, κ κ H˙ = − (ρ + P ) = − φ˙ 2 .(133) 2 φ φ 2 Furthermore, the variation of eq. (129) or the conservation laws of eq. (100) can lead us to the equation of motion of the canonical scalar ﬁeld, which is none else than the Kelin-Gordon equation; beginning with

ρ˙φ + 3H(ρφ + Pφ) = 0 ,

we easily obtain dV φ¨ + 3Hφ˙ + = 0 . (134) dφ In the typical descriptions of inflation (see chapter 1), as generated by such a canonical scalar field, the latter slow-rolls on its potential for as long as it takes for the universe to expand accordingly, and then rolls down t a minimum of the potential, signifying the (graceful) exit from the inflationary era and a reheating phase. As already discussed, this slow-rolling of the scalar field is measured with the use of slow-roll indices e and η, which are defined as

H˙ H¨ e = − and η = − .(135) H2 2HH˙ H˙ It is interesting to note that the deceleration parameter, q = − 1 + is H2 strongly related to them, as

q = e − 1 and q˙ = 2e(e − η) .

Since the universe must be highly accelerating during the inﬂationary era, so q ' −1 and q˙ ' 0, it is easy to see that both slow-roll indices must be positive and really small, e, η 1. This implies that a slow-roll condition exists for the 2.5 viable f (R, φ, X) theories 57 potential and the kinetic energy of the scalar ﬁeld, that eventually the former must highly exceed the latter,

1 φ˙ 2 V(φ) .(136) 2 Thus, the FRW equations are simpliﬁed to κ H2 ' V(φ) and H˙ ' 0 , (137) 3 eventually meaning that the Hubble rate is roughly constant and depends almost entirely on the potential. The slow-roll indices are then expressed in terms of the scalar ﬁeld and its potential as

3φ˙ 2 3φ˙ 2 e = and η ' ,(138) 2V(φ) V(φ) or 2e = η. The Klein-Gordon equation, alng with the second slow-roll condition (η 1) imply that dV 3Hφ˙ ' dφ or √ Z 3 2 Z V(φ) ' −3 (Hφ˙)dφ ' κ φ˙ 2dφ . 2 Subsequently, the slow-roll indices are rewritten as

1 1 dV 2 1 1 d2V e ' and η ' .(139) 2κ V(φ) dφ 2κ V(φ) dφ2

Frome these, we can calculate the spectral index of primordial scalar curvature perturbations, nS, if the Klein-Gordon equation is interpreted as a Bessel differential equation, nS ' 1 − 6e + 2η ,(140) and the tensor-to-scalar ratio, r, considering the magnitudes of the scalar and tensor curvature perturbations, r ' 16e .(141) We should, however, notice that these are not general cases, but refer merely to the linearised slow-roll conditions. Finally, we should also notice that for the Universe to exit the inﬂation, two prerequisites are needed: (1) the scalar ﬁeld must reach the minimum of the potential, and (2) the universe to has reached the desired size. These are possible only if the slow-roll index becomes of the order e ∼ O(1); in other words, only if the spectral index approaches (but not reaches) unity, and the tensor-to-scalar ratio becoming small (certainly below 0.1). 58 the f (r) theory of gravity

2.5.2 Non-canonical scalar ﬁeld inﬂation

The case of a non-canonical scalar field refers to (1) the presence of typical matter fields, aside from the hypothetical fluid representing the scalar field, and (2) the multiplication of the kinetic term of the scalar field with some Brans-Dicke parameters, ω(φ). Keeping everything else as above, the new action becomes

Z R S = d4xp−g + L + L = 2κ φ matter (142) Z p 12H2 + 6H˙ 1 = d4x −g − ω(φ)gµν∂ φ∂ φ − V(φ) + L . 2κ 2 µ ν matter Here, the the slow-roll indices are related to the Brans-Dicke parameter and must be redfined. Generally, one can transform from the non-canonical sclar field, φ, to a canonical one, ϕ, as Z φ q ϕ = |ω(φ0)|dφ0 ,(143) 0 so the Brans-Dicke parameter shall be absorbed by the kinetic term. However, this case is not always possible, due to the fact that the function ϕ = ϕ(φ) must be invertible; furthermore, if ω(φ) < 0, then the scalar field φ is a phantom field and the total energy density is bounded from below, introducing one more undesirable effect during the transformation. In the similar manner to the previous case, the energy-momentum tensors are given as √ ( ) δ −gLφ 1 T φ = = ω(φ)∂ φ∂ φ − g gκλω(φ)∂ φ∂ φ − V(φ) ,(144) µν δgµν µ ν µν 2 κ λ √ ( ) δ ( −gLmatter) T matter = ,(145) µν δgµν

with total mass-energy density

( ) ( ) 1 ρ = T matter + T φ = ρ + ω(φ)φ˙ 2 + V(φ) (146) 00 00 matter 2 and total pressure

1 ( ) ( ) 1 P = δij T matter + T φ = P + ω(φ)φ˙ 2 − V(φ) .(147) 3 ij ij matter 2 The Friedmann-Robertson-Walker equations become

κ κ 1 H2 = ρ = ρ + ω(φ)φ˙ 2 + V(φ) ,(148) 3 3 matter 2 κ κ H˙ = − (ρ + P) = − ρ + P + ω(φ)φ˙ 2 .(149) 2 2 matter matter 2.5 viable f (R, φ, X) theories 59

As for the Klein-Gordon equation, it can again be derived from the conservation laws and take the form 1 dV φ¨ + 3Hφ˙ + = 0 , (150) ω(φ) dφ proposing that ω(φ) 6= 0, a reasonable demand since ω(φ) = 0 would lead to a vanishing kinematic term. Demanding that the slow-roll indices, e and η are defined in the same way as in eq. (135), and that the slow-roll condition holds (e, η 1), then we are lead to 1 dV ω(φ)φ˙ 2 V(φ) and 3Hω(φ)φ˙ ' − ,(151) 2 dφ so the FRW equations are rewritten as κ κ H2 ' (ρ + V(φ)) and H˙ = − ρ + P + ω(φ)φ˙ 2 , 3 matter 2 matter matter in which the Hubble rate is not constant. As a result, in the case the typical matter fields are absent, the slow-roll indices take the form 1 1 1 dV 2 1 1 dω(φ) dV 1 d2V e = and η = − − . 2κ ω(φ) V(φ) dφ κ 2ω(φ)2V(φ) dω dφ ω(φ)V(φ) dφ2 (152) The spectral index and the tensor-to-scalar ratio can be defined as in the case of the canonical scalar field.

2.5.3 Inﬂation with f (R, φ) theories of gravity

A general class of inflationary models can be described by means of the following action, that contains both a non-canonical scalar field and a coupling of this field with scalar curvature through a function, f (R, φ),

Z p 1 1 S = d4x −g f (R, φ) − ω(φ)gµν∂ φ∂ φ − V(φ) + L ,(153) 2κ 2 µ ν matter ∂ f where F(R, φ) = as usual; this action contains the afore-mentioned case of ∂R canonical and non-canonical scalar fields added to an Einstein-Hilbert action, as well as the f (R, φ) = f (φ)R subcases of the theory, where te scalar field is minimally or non-minimally coupled to curvature. The variation of the action with respect to the metric easily yields the following field equations

1 κλ F(R, φ)Rµν − f (R, φ)gµν + gµνd ∇κ∇λF(R, φ) − ∇µ∇νF(R, φ) = 2 (154) κλ (matter) = κ ω(φ) gµνg ∂κφ∂λφ + ∂µφ∂νφ − V(φ)gµν + Tµν

Assuming a ﬂat Friedmann-Lemaître-Robertson-Walker space-time, with the usual metric of eq. (105), we can split the temporal and spatial components of the eq. 60 the f (r) theory of gravity

(154) and obtain the Friedmann equations (see [283, 289]). The ﬁrst one is obtained from the temporal component as 1 1 ( f (R, φ) − RF(R, φ)) + 3H2F(R, φ) − 3HF˙(R, φ) = κ ω(φ)φ˙ 2 + V(φ) − ρ , 2 2 (155) while the second from the temporal components, as

F¨(R, φ) − HF˙(R, φ) + 2HF˙ (R, φ) = −ω(φ)φ˙ 2 .(156)

The variation of the action (eq. (153) with respect to the scalar field φ yields the equation of motion for the respective field, also acting as a continuity equation for the field, 1 ∂ f ∂V φ¨ + 3Hφ˙ + ω˙ (φ)ω˙ − + 2 = 0 . (157) 2ω(φ) ∂φ ∂φ In order to acquire the inflationary dynamics of this model, we derive again from the generalised approach by [197] the following four slow-roll indices

H˙ φ¨ F˙(R) E(˙R) ε = − , ε = , ε = and ε = (158) 1 H2 2 Hφ˙ 3 2HF(R) 4 2HE(R) where 3F˙(R)2 E(R) = F(R)ω(φ) + . 2κφ˙ 2 In addition, we deﬁne the following function, E(R) = ˙ 2 Qs φ 2 2 , H F(R)(1 + ε3) which plays an important role for the calculation of the scalar-to-tensor ratio. Imposing the slow-roll condition, we demand that the slow-roll indices are very small (εi 1) and that

ω(φ˙ )φ˙ ω˙ (φ)φ˙ 2 2ω(φ)φε˙ 1 and 3 ,(159) H2F˙(R, φ) H2F˙(R, φ) HF˙(R, φ) so that the FRW equations (155) and (156) are rewritten as

3H2F(R, φ) RF(R, φ) − f (R, φ) ' V(φ) + κ 2 while the contunuity equation for the scalar ﬁeld becomes 1 1 ∂ f ∂V 3Hφ˙ + ω˙ (φ)φ˙ − = 0 . 2ω(φ) κ ∂φ 2 ∂φ It is proved that following this approximation, where the slow-roll indices are small, the spectral index of primoridal curvature perturbations turns to be [195, 283, 287] nS ' 1 − 4ε1 − 2ε2 + 2ε3 − 4ε4 .(160) 2.6 acosmologicalmodel 61

In the same manner, the tensor-to-scalar ratio is given, in terms of the function Qs, as Q r = 8κ s .(161) F(R, φ)

The function Qs contains the information about the specific form of the f (R, φ) function, and hence the way the scalar field is introduced in the action with respect to the curvature; for example, in the case of a canonical scalar field in the Einstein-Hilbert action that was presented before, this function becomes

φ˙ 2 Q = ,(162) s H2 in the case of the f (φ)R subcases, it becomes

φ˙ 2 3 f˙(φ)2 H f˙(φ) 2H˙ f (φ) Q = + ' − ,(163) s H2 2κH2 f (φ) κH2 κH2 and ﬁnally, in the case of a pure f (R) action, where the scalar ﬁeld vanishes, the results of the previous section are restored by means of

3F˙(R)2 Q = .(164) s 2κH2F(R)

2.6 a cosmological model

e Let us now recall that f (R) = R + f0 + f1R and the Hubble rate contains a singularity at t = ts, h = 0 H β ,(165) (ts − t) where h0 and β are arbitrary constants, suitably chosen so that the Hubble rate will be real and compliable with the observe behaviour of the Hubble rate. Given that R = 6H˙ + 12H2, it is easy to calculate the scalar curvature with respect to time as  12h2  0 whenβ > 1  (t − t)2β  s 12h2 − 6h R = 0 0 whenβ = 1 .(166) ( − )2  ts t  6h − 0 <  β+1 whenβ 1 (ts − t) The above curvature, for different values of β corresponds to different behaviour of the 3-d space, and thus it leads to different types of singularities, as explained in chapter 1. From eqs. (123) and (124) and substituting the scalar curvature of eq. (166), two solutions appear to be consistent. The ﬁrst follows from β = 1 and e > 1 (but e 6= 2), corresponding to the “Big Rip” type of singularity, for h0 > 0 and t < ts, 62 the f (r) theory of gravity

or to the classical type of “Big Bang” singularity, for h0 < 0 and t > ts; the second e follows from β = − , given that e < 1 and −1 < β < 1, which corresponds e − 2 to the “Sudden Future” type of singularity. Notably, if e = 2, then we are drawn to the Starobinsky model -where f (R) ∼ R2- and no singularity exists. From these, we realise that the introduction of R2 in the action in the form of f (R) = R + R2 f˜(R) ,(167) where lim f˜(R) = c1 and lim f˜(R) = c2 , R→0 R→∞ we can assume that f (R) behaves like f (R) ∼ R + R2 for both weak and strong curvature -that is for both early and late times- and the finite-time future singularities disappear. According to refs. [1, 44, 85, 300], the addition of such term in the Einstein-Hilbert action produces an f (R) theory of gravity that realizes viable cosmological models without the occurence of any finite-time singularity; furthermore, this term, as being consistent with ref. [380], generates the early- time accelerating expansion simultaneously. Other phenomenological problems are also solved with the inclusion of such a term (see refs. [23, 225, 301, 385]). Consequently, the f (R) theories that contain such a term, or equivalently, the fulfill the following conditions, generate viable cosmological models, where the early- and late-time accelerations are unified and the future “doomsdays” are avoided. 1. The Einstein-Hilbert action must be retrieved in the weak curvature limit, so f (R) 1 lim f (R) = R and lim = .(168) R→0 R→0 R2 R As a corollary of this, the flat (Minkowski) space-time must always be a solution to the generalised f (R) theory. 2. It is proved (see ref. [136] inter alia) that the deSitter expansion must also be a stable solution of the theory, representing the late-time acceleration. As a result, as the curvature approaches small but non-zero values R ∼ RL ∼ (10−33 eV)2 for the case of late-time acceleration , the following must hold true f (R) = f − f (R − R )2l+2 + O (R − R )2l+2 ,(169) R 0L 1L L L where f0L and f1L are positive constants and l is a positive integer. 3. It is also proved (see refs. [47, 304] for example) that the quasi-deSitter evolution, corresponding to the early-time acceleration, must be a stable solution to the theory. Given that curvature at the very early Universe was large 16∼19 2 R ∼ RI ∼ (10 GeV) , the deSitter solution should not exactly stable but the curvature should decrease very slowly, revealing the condition f (R) = f − f (R − R )2m+2 + O (R − R )2m+2 ,(170) R2 0I 1I I I 2.6 acosmologicalmodel 63

where f0I and f1I are positive constants and m is a positive integer.

4. As we stated, the R2 term should be present in the strong curvature limit, so that the future singularities can be avoided. The imposed condition reads

2−e f (R) lim f (R) = f∞R and lim = f∞ ,(171) R→∞ R→∞ R2−e

where f∞ is a positive and sufﬁciently small constant and 0 ≤ e < 1 -if e = 0, the Starobinsky model is retrieved for large curvatures.

5. In order to avoid the antigravity regime, F(R) > 0 must hold true for all (positive) values of curvature. This condition can be rewritten as

d f (R) 2 ln > − .(172) dR R R

6. Combining conditions (1) and (5), we reach to the conclusion that only positive values of the f (R) are allowed, for positive values of curvature,

f (R) > 0 . (173)

THEGAUSS-BONNETTHEORIESOFGRAVITY 3

The f (G) theory, or the f (R, G) theory of gravity results from the consideration of the Gauss-Bonnet invariant as a higher-order term of curvature that is introduced in the Lagrangian. Usual utilizations of this kind of modified gravity seek to explain the late-time accelerating expansion (see refs. [137, 236, 294, 306–308, 357] and prior to that [100, 199, 256]), since the geometric term seems to play the role of a gravitational Dark Energy or Cosmological constant, or as an alternative to simple [19, 369, 397] or scalarized black holes [40, 116, 131, 271, 368, 409]. Its cosmological influence, however, has been vast; inflationary models have been proposed (see ref. [246] and more recentrly [37, 71, 98, 122, 149, 150, 208, 321, 328, 410, 411, 415]), although they were not considered as serious alternatives to the respective f (R) attempts. The comsological interest lies usually with the canonical dark energy [20, 37, 89, 132, 135, 164, 165, 213, 284], the holographic dark energy [260, 358] or the quintessential dark energy [366]. Generlised models [143, 151, 158, 166, 335] and unified models [290, 322, 384] were currently presented, with viable conditions being produced (see refs. [21, 49, 54, 155, 231, 286, 416] for theoretical and observational constraints). The main stream behind the construction of these models is the Lovelock theory, where higher-order curvature terms are taken into consideration in the action, whereas the Ricci scalar -being the first-order term- leads to the simpler form, the Einstein-Hilbert action. The necessity for their introduction lies with the fact that higher-order terms of curvature may constitute corrections attributed to the quantum or the string theory. As with the Ricci scalar, any higher-order term must be scalar and act like an integral of an invariant.

3.1 general properties

To obtain such an invariant, we are reminded of the Gauss-Bonnet theorem and its generalisation by Chern (see refs. [270, 371] and [239, 388] for some brief discussion on the topic). Given an Euler-Poincarè characteristic on a manifold M, namely a topological invariant of the manifold, χ(M), derived from an Euler class, E(Ω) as Z χ(M) = E(Ω) ,(174) M we can generalise the notion of geodesic curvature ﬂowing through a closed region of the manifold, with boundary ∂M. In this sence, global topology of the manifold, expressed by the Euler-Poincarè characteristics, is linked to local analytical qunatities, namely the Gaussian or the Riemannian curvature. Subsequently,

65 66 the gauss-bonnet theories of gravity

any closed region of a manifold with dimension n > 2 can be patched beginning from an analytically described point towards its totality; furthermore, the curvature of this region, derived on the boundary of it, can be generalised and attain the behaviour of a global quantity for the manifold [233]. More specifically, we can always define a geometric element, G, that fulfills the prerequisites for an Euler class, and obtain the term Z dDxG M which must be a topological surface term for D = n (the dimension of the manifold on which it is defined), or a trivial term for n > D. Given a 4-d metric (Riemannian) manifold -that relates to the 4-d space-time of General Relativity- this term is found to be

2 µν κλµν G = R − 4R Rµν + R Rκλµν .(175) √ This is the Gauss-Bonnet term, that multiplied with −g stands for a topological invariant -and hence, usually called Gauss-Bonnet invariant. The integral of this term on (any region of) the manifold can be generalised to Z dDx f (F) , M where f (G) any arbitrary function of the Gauss-Bonnet term, proposing it is analytic, continuous and differentiable on the manifold. This idea is introduced in the context of gravity theories, with the simpler case being the straight f (G) theory, with action in the form Z 4 p SGB = d x −g f (G) + Lmatter .(176)

Varying this with respect to the metric, gαβ, we reach

2 µν κλµν µν κλµν δG = δ R − 4R Rµν + R Rκλµν = 2RδR − 8RµνδR + 2RκλµνδR ,

where

κ κ κ δR λµν =∇µ (δΓ νλ) − ∇ν δΓ µλ = 1 = gκα ∇ − ∇ (δg ) + ∇ (δg ) + ∇ (δg ) − 2 µ α νλ ν λα λ να − ∇ν − ∇α δgµλ + ∇µ (δgλα) + ∇λ δgµα ,

µν µα νβ ρ ρ δR =g g ∇ρ δΓ αβ − ∇β δΓ ρα = 1 = gκα ∇ − ∇ (δg ) + ∇ (δg ) + ∇ (δg ) − 2 µ α νκ ν κα κ να − ∇ν − ∇α δgµκ + ∇µ (δgκα) + ∇κ δgµα and µν µν µν µν κ µκ ν δR =Rµνδg + gµνδR = Rµνδg + ∇κ g δΓ µν − g δΓ µν , 3.2 field equations and conservation laws 67

and eventually µν κµ λν κλ µν δG =2R − R δgµν + g g ∇κ∇λδgµν − g ∇κ∇λ(g δgµν) + ρσ µν ρµ νσ ρν µσ + 8R Rµρνσδg − 4 R g ∇ρ∇σ + R g ∇ρ∇σ δgµν+ µν κλ ρσ µν + 4R g ∇κ∇λδgµν + 4R ∇ρσ g δgµν − µρκλ ν µρνσ − 2R R ρκλδgµν − 4R ∇ρ∇σδgµν obtaining the following ﬁeld equations µν µ λν µλαβ ν µκνλ µν 2RR − 4R λR + 2R R λαβ − 4R Rκλ = κT ,(177) δL where T = matter the energy-momentum tensor, as in the previous chapters. µν ∂gµν However, the most interesting and usual form is that of the Lovelock theory, with action Z 4 p SGB = d x −g a0 + a1R + a2G + O(a3) + Lmatter ,(178)

where ai are arbitrary real constants, acting as coupling constants [86, 376]. Noting √ 2 µν κλµν the topological invariance of −g R − 4R Rµν + R Rκλµν , and cutting the higher-order Lovelock terms, we may write the simpler action, Z p γ S = d4x −g αRµνR − βR2 + R + L .(179) GB µν 2κ matter The equations of motion for this theory are obtained in the similar manner, varying with respect to the metric; as a result, we have 1 1 α − R Rκλg − ∇ ∇ R + 2R Rαβ + g gκλ∇ ∇ R + gκλ∇ ∇ R + 2 κλ µν µ ν µανβ 2 µν κ λ κ λ µν 1 γ 1 + β R2g − 2RR − 2∇ ∇ R + 2g gκλ∇ ∇ R + R − Rg = T . 2 µν µν µ ν µν κ λ κ µν 2 µν µν (180) The Gauss-Bonnet theory of gravity, as well as the Starobinsky model, and subsequently the Lovelock theory, contain quadratic terms of curvature. This geometric characteristic does not merely extends the Einstein-Hilbert action, in favour of more topological invariants included, but it also takes account of quantum or string-theory corrections. The Gauss-Bonnet, modiﬁed Gauss-Bonnet and “mimetic” Gauss-Bonnet theories -that are presented afterwards- are considered as string-theory-inspired extension of General Relativity, and thus they were considered candidates for a classical large-scale approximation of a uniﬁed theory.

3.2 field equations and conservation laws

Our analysis draws heavily on refs. [45, 135] and [310]. We shall deal with the case of f (R, G) theory, in a generalised way; in order to simplify to R + f (G) Einstein- Gauss-Bonnet theory, one can see ref. [289], as for f (G) pure Gauss-Bonnet theory, [292] gives the special case. 68 the gauss-bonnet theories of gravity

The generalised action for the f (R, G) modiﬁed theory is given as

Z p 1 S = d4x −g f (R, G) + L .(181) modGB 2κ matter To derive the ﬁeld equations all we have to do is vary with respect to the metric; doing so, we introduce ourselves to the complexity of the f (R, G) function, that is a function of two variables. Given the ﬁeld equations as

κλ 1 f (R, G) − 4g ∇ ∇ fG (R, G) R − Rg − R κ λ µν 2 µν 1 − g f (R, G) − f (R, G)R − ∇ ∇ f (R, G) + g gκλ∇ ∇ f (R, G)+ 2 µν R µ ν R µν κ λ R ρ λρσ ακ αλ + RRµν − 4RµρRν + 2Rµ Rνλρσ − 4g g RµανβRκλ fG (R, G)− α α − 2 ∇µ∇ν fG (R, G) R + 4 ∇α∇µ fG (R, G) Rν + ∇α∇ν fG (R, G) Rµ − κλ ακ βλ − 4 gµνR − g g Rµανβ ∇κ∇λ fG (R, G) =

= κTµν , (182)

∂ f ∂ f where f (R, G) = and fG (R, G) = the partial derivatives with respect R ∂R ∂G to the Ricci scalar and the Gauss-Bonnet invariant, we see that they are partial differential equations of the respective function, and consequently they are harder and more arbitrary to solve in order to reconstruct the speciﬁc gravity model. This is clearly demonstrated if we consider a ﬂat FLRW background with metric given from eq. (105) and obtain the Friedmann and Raychaudhuri equations as 1 h 1 ρe f f = ρ + fR(R, G)R − f (R, G) − 6H f˙R(R, G)+ (183) fR(R, G) 2κ 3 i + G fG (R, G) − 24H f˙G (R, G) (184) 1 h 1 Pe f f = P + fR(R, G)R − f (R, G) + 4H f˙R(R, G) + 2 f¨R(R, G)− (185) fR(R, G) 2κ 2 2 i − G fG (R, G) + 16H H˙ + H f˙G (R, G) + 8H f¨G (R, G) ,(186)

where ρe f f and Pe f f correspond to eqs. (111) and (111), so they are written with respect to the Hubble rate (see ref. [45]). However, we know that the coupling of the Ricci scalar and the Gauss-Bonnet invariant is very important for the form of the f (R, G) function; specifically, if the two geometric variables can be separated in a specific way, the field equations are further simplified and may be separated as well. We will consider two case of coupling, hence separation, of the Gauss-Bonnet term to the Ricci scalar:

1. Minimal coupling, where f (R, G) = α1 f1(R) + α2 f2(G), so the two curvature terms affect geometry via different mechanisms. 3.2 field equations and conservation laws 69

2. Non-minimal coupling, where f (R, G) = f1(R) f2(G), so the two curvature terms do not act independently.

3.2.1 Minimal coupling

As analysed in [310], the addition of f1(R) to f2(G) leads to the following action

Z p 1 S = d4x −g α f (R) + α f (G) + L ,(187) GB 2κ 1 1 2 2 matter where the effects of the two geometric terms, such as the possible presence of ghost modes, can be identiﬁed from one another. Henceforth, we shall consider a1 = a2 = 1 in order to simplify the relationships, without any lack of generality. Varying the action (187) with respect to the metric, gαβ, we obtain the general ﬁeld equations

2κ h (matter) 1 µ (c)i Gαβ = Tαβ + f1(R) − RF1(R) gαβ + ∇β∇αF1(R) − ∇ ∇µF1(R)gαβ + Tαβ , F1(R) 2 (188) ∂ f where F (R) = 1 , 1 ∂R 1 G = R − Rg αβ αβ 2 αβ the Einstein tensor, √ ( ) 2 ∂ ( −gL ) T matter = −√ matter αβ −g ∂gαβ the stress-energy-momentum tensor for the matter ﬁelds, and

( ) 1 T c = g f (G) − 2RR − 8RµνR + 2R R κλµF (G)+ αβ 2 αβ 2 αβ αµβν ακλµ β 2 µ µ µ + 2 ∇α∇β − gαβ∇ ∇µ RF2(G) − 4∇α∇µ R βF2(G) + 4∇ ∇µ RαβF2(G) + µν µ ν + 4gαβ∇µ∇ν R F2(G) − 4∇µ∇ν Rα β F2(G) ,

∂ f where F (G) = 2 , an “energy-momentum” tensor for the Gauss-Bonnet term, 2 ∂G acting like correction to the relativistic field equations (adapted from eqs. (94-94), p. 12 of ref. [197]). It should be noted that eq. (188) does not contain any third- or higher-derivative terms of curvature, so the action of eq. (187) is non-degenerate and respects the Ostrogradski theorem, produces field equations up to second- order derivatives. As for the conservation laws, we know from refs. [337, 365] inter alia that the Nöther symmetries are preserved in a modified Gauss-Bonnet theory; thus, there are conserved quantities and corresponding conservation laws, in the same manner as in the f (R) theories. So, we can apply the formula from ref. [88] and 70 the gauss-bonnet theories of gravity

obtain the “energy” conditions necessary for the conservation of energy and mo- αβ (matter) mentum to preserve their original form, ∇αT = 0. It is easy to see that the following relationship must hold for the covariant derivative of the “energy- momentum” tensor of the curvature corrections, 1 ∇ Tαβ (c) = ∇ Gαβ − κTαβ (matter) − gαβ ∇ f (R) − R∇ F (R) − F (R)∇ R . β β 2 β 1 β 1 1 β (189) Given the ﬂat FLRW metric, we can transform the general ﬁeld equations to simpler scalar equations, the Friedmann and Raychaudhuri equations. Obtaining from eq. (105)

2 2 4 2 Ri0j0 = −a(t) H˙ + H δij , Rijkl = a(t) H δikδjl − δilδjk , 2 2 2 R00 = −3 H˙ + H , Rij = a(t) H˙ + 3H δij , R = 6H˙ + 12H2 and G = 24H2 H˙ + H2 ,

we can easily reach the following equations,

1 h 1 ρ = ρ + RF (R) − f (R) − 6HF˙ (R)+ e f f ( ) matter 1 1 1 F1 R 2κ (190) 3 i + GF2(G) − f2(G) − 24H F˙2(G)

and 1 h 1 Pe f f = Pmatter + f1(R) − RF1(R) + 4HF˙1(R) + 2F¨2(R)+ F1(R) 2κ 2 2 i + f2(G) − GF2(G) + 16H(H˙ + H )F˙2(G) + 8H F¨2(G) . (191)

3.2.2 Non-minimal coupling

Although the case of a scalar field coupled to the Gauss-Bonnet term has been issued quite rigorously (see for example refs. [45, 53, 121, 135, 164–166, 197, 206– 208, 290, 308, 322]) the case of non-minimal coupling of the Gauss-Bonnet term to the Ricci scalar has not been studied. We shall briefly present this case, despite its complexity and minimum interest in the respective literature. The action for non-minimal coupling of the Ricci scalar and the Gauss-Bonnet term reads Z p 1 S = d4x −g f (R) f (G) + L ,(192) GB 2κ 1 2 matter 3.2 field equations and conservation laws 71 where the effects of the two geometric terms cannot be separated from one another. Varying the action (192) with respect to the metric, gαβ, we obtain the general field equations

κλ Gαβ F1(R) f2(G) − 4g ∇κλ f1(R)F2(G) =

h (matter) µ µρσ µκ νλ =2κ Tαβ + f1(R)F2(G) RRαβ − 4RαµRβ − 4Rα Rβµρσ − 4g g RαµβνRκλ + G G R R R+G i + f1(R)Tαβ + F1(R)tαβ + f2(G)tαβ + F2(G)Tαβ + Sαβ , (193)

∂ f ∂ f where F (R) = 1 and F (G) = 2 , 1 ∂R 2 ∂G 1 G = R − Rg αβ αβ 2 αβ the Einstein tensor, √ ( ) 2 ∂ ( −gL ) T matter = −√ matter αβ −g ∂gαβ the stress-energy-momentum tensor for the matter ﬁelds and

(R) = ∇ ∇ ( ) − µ∇ ∇ ( ) + ( κλ + µκ νλ )∇ ∇ ( ) Tαβ 2R α β f1 R 8R(α β) µ f1 R 4 gαβR g g Rαµβν κ λ f1 R (R) κλ tαβ =∇α∇βF1(R) + gαβg ∇κ∇λF1(R) (G) = ∇ ∇ (G) − µ∇ ∇ (G) + ( κλ + µκ νλ )∇ ∇ (G) Tαβ 2R α βF2 8R(α β) µF2 4 gαβR g g Rαµβν κ λF2 (G) κλ tαβ =∇α∇β f2(G) + gαβg ∇κ∇λ (R+G) κλ Sαβ =2∇(αF1(R)∇β) f2(G) + 2gαβg ∇(κ F1(R)∇λ) f2(G) + 2R∇(α f1(R)∇β)F2(G)− − µ∇ ( )∇ (G) + ∇ (G)∇ ( )+ 8R(α β) f1 R µF2 β)F2 µ f1 R κλ µκ νλ + 8 gαβR − g g Rαµβν ∇(κ f1(R)∇λ)F2(G) the “energy-momentum” tensors associated with additional curvature terms. As in the case of the minimal coupling, eq. (193) does not contain any third- or higher-derivative terms of curvature, so the action of eq. (192) is non-degenerate and respects the Ostrogradski theorem, produces ﬁeld equations up to second- order derivatives. As for the conservation laws, what we stated about Nöther symmetries in a modiﬁed Gauss-Bonnet theory is still in existence [337, 365]; hence, we can once more apply the formula by [88] and obtain the “energy” conditions necessary for the conservation of energy and momentum to be preserved in their original form, αβ (matter) ∇αT = 0. After some manipulations, the “energy-momentum” tensors 72 the gauss-bonnet theories of gravity

of the curvature corrections must fulﬁll the following relationship in order fot the conservation laws to hold,

αβ (G) αβ (G) αβ (R) αβ (R) αβ (R+G) ∇β f1(R)T + F1(R)t + f2(G)t + F2(G)T + T + αβ αµ β αλρσ β αµβν + ∇β f1(R)F2(G) RR − 4R R µ + 2R R αρσ − 4R Rµν −

αβ κλ − G ∇β F1(R) f2(G) − 4g ∇κ∇λ f1(R)F2(G) . (194)

Given the ﬂat FLRW metric, we can transform the general ﬁeld equations to the simpler equationsFriedmann and Raychaudhuri equations. Obtaining from eq. (105)

2 2 4 2 Ri0j0 = −a(t) H˙ + H δij , Rijkl = a(t) H δikδjl − δilδjk , 2 2 2 R00 = −3 H˙ + H , Rij = a(t) H˙ + 3H δij , R = 6H˙ + 12H2 and G = 24H2 H˙ + H2 ,

we can easily reach the following equations,

ρmatter 1 h RF1(R) − f1(R) F˙1(R) f˙2(G) ρe f f = + − 6H + + F1(R) 2κ F1(R) F1(R) f2(G) 1 3 3 i + G f1(R)F2(G) − 24H f˙1(R)F2(G) − 24H f1(R)F˙2(G) F1(R) f2(G) (195)

and

Pmatter 1 h RF1(R) − f1(R) F˙1(R) f˙2(G) Pe f f = + + 4H + + F1(R) 2κ F1(R) F1(R) f2(G) F¨ (R) F˙ (R) f˙ f¨ (G) + 2 1 + 2 1 2 + 2 − F1(R) F1(R) f2(G) f2(G) f (R)F (G) f˙ (R)F (G) + f (R)F˙ (G) − G 1 2 + 16H2(H˙ + H2) 1 2 1 2 + F1(R) f2(G) F1(R) f2(G) f¨ (R)F (G) + f˙ (R)F˙ (G) + f (R)F¨ (G) i + 8H2 1 2 1 2 1 2 F1(R) f2(G) (196)

3.3 “mimetic” ghost-free theory

It has been noted in ref. [35, 87, 90, 386] that the f (G) theory can be expressed in a mimetic form, by means of Lagrange multipliers. This idea was not proved fruitful, considering the emergence of “ghost modes” that could not be excluded and caused the respective models to disagree with current theoretical beliefs and observational data. In [290], a different formulation was proposed of this “mimetic” 3.3 “mimetic” ghost-free theory 73 f (R, G) theory was proposed that could eliminate the “ghost modes”, by means of the Lagrange multipliers. We proceed by repeating their results in the case of minimal coupling. Using the Bianchi identities, the terms concerning the Gauss-Bonnet invariant are shorted out as

( ) 1 T c = g f (G) − GF (G) + D µν∇ ∇ F (G) ,(197) αβ 2 αβ 2 2 αβ µ ν 2 where µν = µ ν − µν − µρ ν σ − µρ νσ + Dαβ 2 δ(α δβ) gαβ g R 4 2g δ(α δβ) gαβ g g Rρσ (198) µν ρ(µ| σ|ν) + 4Rαβg − 4Rαρβσg g .

Furthermore, they state that a speciﬁc trace of the above coupling tensor is proportional to the Einstein tensor,

1 gκλD µν = −4Gµν = −4 Rµν − Rgµν ,(199) κλ 2 and the following component form

00 00 0 00 00 0 0 D00 = 2R − 2g00g R − 8R 0 + 4g00R + 4g R00 − 4R 0 0 00 00 0 0 00 00 Dij = 4gij R − 4R i j − 2gij g R + 4Rijg .

If a specific gauge is chosen, such that non-diagonal elements of the metric vanish 00 (g0i = gi0 = 0), then the temporal component D00 vanishes, but the spatial com- 00 ponents Dij do not. As a result of this, the general form of the field equations (eq. 188) includes the fourth-order derivative of the metric with respect to the the cosmic time; this fourth-order derivative might generate ghost modes. Specifically, if we consider the variation of the Ricci tensor and the Ricci scalar under the gauges µ µ ∇ δgµν = 0 δg µ = 0 , we shall have

1 µ κ λ κλ µ δR = 2R δg − 2R δg − ∇α∇β g δg − ∇ ∇µ δg αβ 2 (α| ρ|β) α β κλ κλ αβ (200) κλ µ κλ δR = −R δgκλ − ∇ ∇µ g δgκλ , and the corresponding variation of the Gauss-Bonnet invariant

µν ρσ µ ν µν κ δG = − 2RR δgµν + 8R R ρ σδgµν + 4R ∇ ∇κδgµν− 201 κνρσ λ µρνσ ( ) − 2R R νρσδgκλ − 4R ∇ρ∇σδgµν .

We easily see that the variations of the Ricci tensor and the Ricci scalar contain derivatives of the metric with respect to the cosmic time as high as the third-order alone; on the other hand, the variation of the Gauss-Bonnet invariant contains 74 the gauss-bonnet theories of gravity

derivatives of fourth-order, that will be retained under a small linear perturbation of the metric. As a result, the perturbed ﬁeld equations (eq. 188). In [290] it is stated that, the propagating ghost mode is a scalar mode expressed in respect to the Gauss-Bonnet invariant.

In order to avoid the appearance of such terms, generated by the Gauss-Bonnet invariant, we introduce the auxiliary scalar ﬁeld χ = χ(xµ) that is coupled to the Gauss-Bonnet invariant. The action of this f1(R, χ) theory is the following Z p 1 c1 S = d4x −g f (R, χ) − V(χ) − h(χ)G + L ,(202) 2κ 1 2 matter

where V(χ) is the potential of the scalar field and c1 the coupling constant of the scalar field to the Gauss-Bonnet invariant by means of a coupling function h(χ); this can be seen as a specialised case of action in eq. 187, where f2(G) = c − 1 h(χ)G − V(χ), that is a simplified case of the Gauss-Bonnet invariant coupled 2 to an arbitrary function of the scalar field. In this form, a possible coupling of the scalar field, χ, with the Ricci curvature is also considered, again without any lack of generality -since this coupling can be nulled at any moment.

Varying the action (202) with respect to the metric, gαβ, we obtain the general ﬁeld equations

2κ h ( ) 1 G = T matter + f (R, χ) − RF (R, χ) − 2V(χ)g + αβ ( ) αβ 1 1 αβ F1 R, χ 2 (203) µ (c)i + ∇α∇βF1(R, χ) − ∇ ∇µF1(R, χ)gαβ + Tαβ and with respect to the scalar field, χ, we obtain the general equation of motion for the scalar field ∂V ∂ f 1 + 1 + T(c) = 0 , (204) ∂χ ∂χ 2 where ( ) h 1 T c = −c h(χ) Gg + 4R R µ + 4RµνR − 2R µνρR − 2RR − αβ 1 2 αβ αµ β αµβν α βµνρ αβ 1 − 4 ∇µ∇νh(χ)R − ∇µ∇ h(χ)R + 2∇ ∇ h(χ)Rµ − ∇ ∂ h(χ)R + αµβν µ αβ µ (β α) 2 β α µν µ i + 2 ∇µ∇νh(χ)R − ∇ ∇νh(χ)R gαβ and ∂h T(c) = c G 1 ∂χ the tensor and scalar modes corresponding to the modified curvature, due to the Gauss-Bonnet and the scalar field coupling (modified for eqs. (46-47), p. 7 and (93-94), p.12 of [197]). We should notice that the tensor mode is but the previous c (eq. 197), is we assume f (G) = − 1 h(χ)G − V(χ) ; then 2 2 ( ) 1 T c = g f (G) − GF (G) + D µν∇ ∇ F (G) = D µν∇ ∇ h(χ) αβ 2 αβ 2 2 αβ µ ν 2 αβ µ ν 3.3 “mimetic” ghost-free theory 75

Beginning with the equation of motion of the scalar field (eq. 204), ∂ f ∂V c ∂h 1 + + 1 G = 0 , ∂χ ∂χ 2 ∂χ that has solutions of the general form χ = χ(G). Since the scalar field is given as function of the Gauss-Bonnet invariant, the field equations (eq. 203) are fourth- order differential equations of the metric with respect to the cosmic time; such a form of field equations can generate the kind of ghost modes we prior analysed. 1 However, introducing a canonical kinetic term in the form of ∂ χ∂ χ in the eq. 2 α β (203), we may express the field equations of our theory in the following form 1 1 G − f (R, χ) − RF(R, χ) − ∇ ∇ F(R, χ) + g ∇ ∇µF(R, χ) = 2κ αβ 2 α β αβ µ 1 ( ) 1 1 1 = T matter + ∂ χ∂ χ − g ∂ χ∂µχ + V(χ) + D µν∇ ∇ h(χ) 2 αβ 2 α β 2 αβ 2 µ αβ µ ν (205) and ∂ f ∂V c ∂h ∇ ∇µχ = 1 + + 1 G .(206) µ ∂χ ∂χ 2 ∂χ These equations do not contain derivatives of order higher than second; as a result, the above system of equations along with initial conditions for the metric, the scalar field and their temporal evolution (gαβ and g˙αβ, χ and χ˙) are sufficient to yield the solutions. Considering the initial conditions on a spatial hypersurface, the solutions are uniquely defined function of the cosmic time, gαβ = gαβ(t) and χ = χ(t). Thus, the ghost modes produced by the Guass-Bonnet invariant vanish. In a mimetic form, the above equations of motion (eq. 205 and 206) can be obtained by a mimetic gravity, expressed by an action along with a constraint and a Lagrange multiplier field, λ = λ(xµ). The action takes the final form

Z 4 p 1 1 µ 1 c1 S˜ = d4x −g f (R, χ) + λ ∂ χ∂µχ + − ∂ χ∂µχ − V(χ) − h(χ)G + L , 2κ 1 2 µ 2 2 µ 2 matter where µ a constant with mass-dimension one. The variation of the action with respect to the metric yields the field equations (205), the variation with respect to the scalar field yields the equation of motion (206); finally, varying with respect to the Lagrange multiplier field, λ, we obtain the the constraint imposed

1 µ4 ∂ χ∂µχ + = 0 , (207) 2 µ 2 µ4 meaning that the kinetic term becomes a constant, equal to − . As a result, we 2 may redeﬁne the potential of the scalar ﬁeld as an effective potential, containing also the constant kinetic term, 1 µ4 V˜ (χ) = V(χ) + ∂ χ∂µχ = V(χ) − .(208) 2 µ 2 76 the gauss-bonnet theories of gravity

We can also redeﬁne the f1(R, χ) as

2 4 f˜1(R, χ, X) = f1(R, χ) − 2λX = f1(R, χ) + λκ µ ,(209) 1 where X = ∂ χ∂µχ the canonical kinetic form of the scalar field. Thus the action 2 µ is rewritten as Z p 1 ω c1 S˜ = d4x −g f˜ (R, χ, X) − ∂ χ∂µχ − V˜ (χ) − h(χ)G + L . 2κ 1 2 µ 2 matter (210) This form is not chosen accidentally, but because it reminds similar actions used by [197] (eq. (45), p. 6) and [321] (eq. (2), p. 2); in our case, ω(χ) = −λ and we may consider simple cases for f (R, χ), c1 and h(χ). Turning to that, the analysis of [322] and [290], which we follow closely, con- centrates on the simplest case of the f (R, χ) theory, considering f1(R, χ) = R and a flat FRW vacuum background. Although this case might seem oversimplified, it is justifiable and may highlight the interesting sides of this theory.

3.3.1 The Flat FRW Vacuum without ghosts

Let us consider a ﬂat Friedmann-Lemaître-Robertson-Walker space-time, given from the line element of eq. (105), and a torsionless, symmetric and metric compatible connection, namely the Levi-Civitta connection,

a˙ Γ0 = 0 Γ0 = Γ0 = 0 Γ0 = aa˙ δ Γ0 = Γ0 = δ ,(211) 00 i0 0i ij ij i0 0i a ij

where a = a(t) the scale factor and δij the Kronecker tensor. The Ricci scalar is given as R = 6H˙ + 12H2 (212) while the Gauss-Bonnet invariant becomes

G = 24H2(H˙ + H2) ,(213) a˙ where H = the Hubble expansion rate. We should notice that dots stand for a da the derivatives with respect time (a˙ = ), while tones stand for the derivatives dt da with respect to the e-foldings number, N (a0 = ). dN Applying these to the ﬁeld equations (eq. 205) and distinguishing the temporal and the spatial components, we obtain the two Friedman equations, the ﬁrst one

2 ω(χ) R f˜ (R, χ) 3F (R, χ)H ∂h 1 ( ) χ˙ 2 + V˜ (χ) + F (R, χ) − 1 − 1 + 12c χ˙ H3 = − T matter 2 2κ 1 2κ 2κ 1 ∂χ 2 00 (214) 3.3 “mimetic” ghost-free theory 77 and the second one

2 ω(χ) f˜ (R, χ) 3F1(R, χ) H˙ + 3H F˙ (R, χ)H F¨ (R, χ) χ˙ 2 − V˜ (χ) + 1 − + 1 + 1 − 2 2κ 2κ κ2 2κ 2 ∂ h ∂h ∂h 1 ( ) − 4c H2χ˙ + H2χ¨ + 2H(H˙ + H2)χ˙ = T matter , 1 ∂χ2 ∂χ ∂χ 2 ii (215)

(matter) (matter) where T00 = ρ the matter-energy density and Tii = P the pressure. In the same manner, the equations of motion of the ﬁeld (eq. 206) turn into

∂V˜ 1 ∂ω 1 ∂ f˜ ∂h ω(χ)χ¨ + 3ω(χ) χ˙ + χ˙ − + 12c H2(H˙ + H2) = 0 , (216) ∂χ 2 ∂χ 2κ ∂χ 1 ∂χ which is but an augmented form of the Klein-Gordon equation (see also ref. [322]). Following ref. [290], we consider the simplest form of the theory, speciﬁcally 4 f1(R, χ) = R or f˜1(R, χ, X) = R + λκµ , as if in a GR-like case; as a result, ∂ f F (R, χ) = 1, F˙ = F¨ = 0 and 1 = 0. We also consider a simple coupling 1 1 ∂χ (matter) constant, c1 = −2 and a Universe empty of matter ﬁelds (vacuum), Tαβ = 0. Remembering that ω(χ) = −λ, the Friedmann equations are rewritten as

λ 3H2 ∂h χ˙ 2 + µ4 − V˜ (χ) + + 24 χ˙ H3 = 0 (217) 2 2κ ∂χ and λ R 3 H˙ + 3H2 ∂2h ∂h ∂h χ˙ 2 + µ4 + V˜ (χ) − + − 8 H2χ˙ + H2χ¨ + 2H(H˙ + H2)χ˙ = 0 , 2 2κ 2κ ∂χ2 ∂χ ∂χ (218) while the Klein-Gordon equation becomes

∂V ∂λ ∂h λχ¨ + 3λ + χ˙ + 24 H2(H˙ + H2) = 0 . (219) ∂χ ∂χ ∂χ

Following [290], we shall consider both the scalar field and the Lagrange multiplier field as dependent only on the cosmic time, χ = χ(t) and λ = λ(t), since we ae on a FRW background; so the analysis remains on a spatial supersurface. As we can easily see from the constraint (eq. 207), the time evolution of the scalar field is constant, χ˙ = µ2 (220) and the scalar field is a linear function of time,

χ(t) = µ2t .(221) 78 the gauss-bonnet theories of gravity

Using this and eq. (212), we may further simplify the Friedmann equations and the Klein-Gordon equation. Eqs. (217) and (218) are rewritten as

3H2 µ4 1 + λ − V˜ (µ2t) + 12µ4 H3h˙ = 0 and (222) 2κ 2 2 1 1 2H˙ + 3H2 − V˜ (µ2t) + 8µ2 HH˙ + H2h˙ + 4µ4 H2h¨ = 0 , (223) 2κ 2 indicating the temporal evolution of the coupling function h(µ2t); similarly, eq. (219) is rewritten as

µ2λ˙ + 3µ2 Hλ + 24µ2 H2H˙ + H2h˙ − µ2V˜˙ (t) ,(224)

as the temporal evolution of the Lagrange multiplier. From eq. (222), we can solve with respect to λ and obtain

3H2 1 λ = − + V˜ (µ2t) − 24H3h˙ . κ2µ4 µ4

Substituting λ into eq. (224), we end up with end up with eq. (223); if, however, we solve the second Friedmann equation with respect to V˜ (µ2t), we obtain

1 V˜ (µ2t) = 2H˙ + 3H2 + 8µ2 HH˙ + H2h˙ + 8µ4 H2h¨ .(225) κ2 Subsequently, the Lagrange multiplier is given as

2H˙ 8 λ(µ2t) = + H2H˙ − H2h˙ + 8H2h¨ .(226) κ2µ4 µ2

Eqs. (225) and (226) imply that the effective potential and the Lagrange multiplier follow the choice of the coupling function h(χ); in fact, if the function is kept simple (e.g. an exponential or a power-law), the effective potential and the Lagrange multiplier have the very same functional form. As a result, our choice of simple exponential and power-law coupling functions is not incidental, but comes in accordance to the assumptions made for the potential by [321] and the results presented there. Furthermore, [290] state that the above procedure is a reconstruction technique for a model following their action, in our case the action (210). This is the way we also follow; yet, our purpose is not to reconstruct the potential of the scalar ﬁeld χ, but the inﬂationary dynamics such a model supplies.

3.4 a cosmological model

Let us consider the simplest case of a flat FLRW Universe and f (R, G) gravity, given by eqs. (184 and 186) for ρmatter = Pmatter = 0. Despite initial objections [118, 236], cosmological solutions such as this have been proved stable and viable, capable of representing both the early- and late-time accelerations of the Universe (see 3.4 acosmologicalmodel 79 refs. [45, 114, 123, 133, 135, 261, 322] inter alia for some successful reconstructions of viable cosmologies and [289] for a bried summary). Two broad cases and a third will be examined referring to the form of the f (R, G) function, namely on the coupling of the Ricci scalar to the Gauss-Bonnet invariant. The first two reflect the previous analysis on minimal and non-minimal coupling, in other words on the addition or the multiplication of the two geometric features; in both cases a simple power-law will be employed for the curvature corrections apart from the relativistic term. The last one encapsulates the temporal evolution of the f (R, G) function and examines the existence and avoidance of the finite-time future singularities. The analysis focuses on ref. [114, 123, 261, 322] and [45]; it purpose is to explain the main issues encountered in the cosmological models of the modified Gauss-Bonnet theory of gravity.

α β 3.4.1 Minimal coupling: f (R, G) = R + f1R + f2G

Let us consider the case of minimal coupling, so that

α β f (R, G) = R + f1R + f2G ; 1 if f = and α = 2, the Ricci terms correspond to the Starobinski model for 1 6M2 inﬂation, while the Gauss-Bonnet term is completely new. The Friedmann and Raychaudhuri equations (eqs. (190) and (191) respectively) are written as 1 H˙ = α(α − 1) f HRα−2(R¨ − R˙ ) + (α − 2)Rα−3R˙ 2+ α−1 β−2 1 2 + 2α f1R + 8β(β − 1) f2 HG G˙ 2 β−2 β−3 2 + β(β − 1) f2 H G (G¨ − G˙) + (β − 2)HG G˙ and (α − 1) f Rα − 6α(α − 1) f HRα−2R˙ + β f G β H2 = 1 1 2 . α−1 β−2 6 + 6α f2R + 8β(β − 1) f2 HG G˙ According to [123], such a “toy model” in vacuum is capable to produce two types of inﬂation, a Starobinsky R2-led one and a Gauss-Bonnet G-led one, in the case of α = β = 2; the introduction of both the R2 and the G2 terms is equivalent 1 1 with introducing two effective masses, mR = and mG = p respectively, 6 f1 2 3 12β the scale factor of the Universe is given as

√t √ t 6 f 6 96 f a1(t) ∼ e 1 and a2(t) ∼ e 2 , that approach quasi-deSitter expansion of the early Universe. Ref. [322] also arrive to viable inﬂationary models in vacuum, where the Hubble rate becomes   β  t − c1β H(t) = ( − ) ,  2 1 2β  t − 2c2(1 − 2β) 80 the gauss-bonnet theories of gravity

where c1 and c2 are arbitrary integration constants; note that both Hubble rates reach a finite-time singularity in t = c1β and t = 2c2(1 − 2β) respectively. In the same manner, ref. [289] states that the same “toy model” with ideal fluid for a vanishing Starobinsky term ( f1 = 0), so that inflationary solutions are driven out, can generate a viable late-time accelerating expansion. For this, the small curvature regime is assumed and the scale factor is found to be  h a0t 0 , when h0 > 0 a(t) = , h a0(ts − t) 0 , when h0 < 0

where

− 1 h f2(β − 1) 3 β 3(1+w) 4β a0 = − 24|h0(h0 − 1)| (h0 − 1 + 4β) and h0 = . ρ0(h0 − 1) 3(1 + w)

Here a “Big Rip” future singularity may occur in t = ts, depending highly on the values of β, since the effective equation of state yields

2 1 + w we f f = −1 + = −1 + . 3h0 2β

1 Given β < 0, the effective equation of state yields we f f < −1, indicating a phantom evolution with negative h0 -even if we f f > −1, the evolution shall remain phantom. Near the “Big Rip” singularity, however, the Einstein curvature becomes large and dominates the Gauss-Bonnet term, so the latter can be neglected; in that case, the Universe behaves like matter- or radiation- dominated, and the singularity is avoided. Therefore, the phantom era is transient. 1 2 Given 0 < β < , the Gauss-Bonnet term is again neglected, since the Ein- 2 stein curvature becomes again large enough for the small curvature condi- −2 tion not to hold. Hence, the Universe is matter-dominated with ρmatter ∼ t , as in General Relativity, and G ∼ t−4. At late times, however, the Gauss- Bonnet term may become large gain and no more neglected, so the Uni- verse is bound to enter an asymptotically deSitter phase -even is it starts 1 from decelerating eras (where w > − ). Consequently, a transition from 3 matter-dominated decelerating expansion to accelerating expansion in possible within this context.

α β 3.4.2 Non-minimal coupling: f (R, G) = f0R G

Let us consider the case of non-minimal coupling, so that

α β f (R, G) = f0R G , 3.4 acosmologicalmodel 81 that is associated with the Nöther symmetries if β = 1 − α. If α = 2, then β = −1. The Friedmann and Raychaudhuri equations (eqs. (190) and (191) respectively) are written as 1 H˙ = × α−1 β α−1 β−1 α β−2 2α f0R G + 8β f0 αR G G˙ + (β − 1)R G G˙ α−2 β α−1 β−1 f0αH (α − 1)R G R˙ − βR G G˙ − α−3 β α−2 β−1 α−1 β−2 − α f0 (α − 1)(α − 2)R G R¨ + 2β(α − 1)R G R˙ G˙ + β(β − 1)R G G¨ + 3 α−1 β−1 α β−2 + 4β f0 H αR G R˙ + (β − 1)R G G˙ + 2 α−2 β−1 α−1 β−2 α β−3 + 4β f0 H α(α − 1)R G R¨ + 2α(β − 1)R G R˙ G˙ + (β − 1)R G G¨ and

α−1 β α−1 β−1 α β−2 6α f0R G + 8β f0 H αR G R˙ + (β − 1)R G G˙ H2 = . α β α−2 β α−1 β−1 (a − 1 + β) f0R G + 6α f0 H (α − 1)R G R˙ + βR G G˙

According to [123], such a “toy model” in vacuum is capable to produce early- time accelerating expansion, a Starobinsky R2-led one and a Gauss-Bonnet G-led one, in the case of α = β = 2; the introduction of both the R2 and the G2 terms is equivalent with a scale factor in the form

z a(t) = a0t , where l(2 − 2α − α2) z = 2α − 1 or z = √ 1 , 617 − 12α − 3α2 − α3 + 3 33 − 48α + 6α2 + 6α3 + 3α4 3 with   1 l = √ 1 ± i 3  . 2 The spectral index and the tensor-to-scalar ratio obtained for this “toy-mode” agree with observations by Planck and BICEP2/KeckArray [6]. It is notable that no ﬁnite-time singularity occurs in the case of non-minimal coupling.

3.4.3 The ﬁnite-time singularities: f (R, G) = R + p(t)G + q(t)

The last case considered is due to ref. [45] and can be considered an extension of the minimal coupling; the key difference is that the coefficients of coupling are not constant but time-varying, so as to exogenously contain the finite-time singularities. Specifically, it is assumed that

f (R, G) = R + p(t)G + q(t) , 82 the gauss-bonnet theories of gravity

where p(t) and q(t) are unknown functions of time, used in order to realise the ﬁnite-time singularities at some moment ts. Varying the corresponding action with respect to the metric and assuming a ﬂat FLRW space-time, we obtain the Friedmann and Raychaudhuri equations in the form 1 12 ρ = ρ − q(t) − H3 p˙(t) , e f f 2κ κ 1 8 P = P − q(t) + H(H˙ + H2)p˙(t) + d f rac4κH2 p¨(t) , e f f 2κ κ 3 1 where ρ = H2 and P = − (3H2 + 2H˙ ). Varying with respect to the coor- e f f κ e f f κ dinate time, we obtain a constraint for the p(t) and q(t) functions in the form

p˙(t)G + q˙(t) = 0 .

Combining the constraint with the Friedmann equations, and remembering that G = 24H2(H˙ + H2), we may arrive to the following two differential equations for the q(t) function,

H˙ + H2 H˙ + H2 3 q˙(t) − q(t) = 2κ H2 − ρ , H H κ 2(H˙ + H2) 1 q¨(t) + q¨(t) + 6(H˙ + H2)q(t) = 8κ(H˙ + H2) P + (3H2 + 2H˙ ) . H 2κ From these differential equations, supposing that we deal with speciﬁc matter ﬁelds and Hubble rate, we can solve and obtain the q(t) function in closed form; utilizing the constraint, we can also obtain the p(t) function in closed for via the integral 1 Z q˙(t) p(t) = − dt . 24 H2(H˙ + H2) However, we need to notice that the differential equations for q(t) come along with a pole in H(t) = 0. In this case, a static Universe is not a viable solution, and only decelerating or accelerating expansion can come as possible solutions. Afterwars, [45] suppose a Hubble rate in the form h ( ) = 0 H t β ,(227) (ts − t)

where h0 and β are real constants. The scalar curvature with respect to time is given by eq. (50) and the Gauss-Bonnet term as

 24h4 ∼ 0 whenβ > 1  (t − t)4β  s 24(h3 + h3) G = 24H2 H˙ + H2 = 0 0 whenβ = 1 .(228) ( − )4  ts t  24h β ∼ 0 <  3β+1 whenβ 1 (ts − t) 3.4 acosmologicalmodel 83

Consequently, we may discuss each case separately, understand the emerging singularities and realise some R + f (G) model, straightly derived from this sup- position, that either contains or avoids the singularities.

1 Type I singularity when β = 1. Here, the functions p(t) and q(t) can be derived analytically as

3−h0 1 (ts − t) p(t) = (2ts − t) t + q0 + p0 , 4h0(h0 − 1) 3 − h0 3 ts−t 2−h 2 − ( − ) 0 6h 24h0 2h (h −1) q0 ts t ( ) = − 0 − 0 0 q t 2 3 , (ts − t) (ts − t)

where p0 and q0 are integration constants. These may hold true only if h0 6= 0, 1, 3.1 From the constraint, we may derive the coordinate time as a function of the Gauss-Bonnet,

1 ! 4 24 h3 + h4 t = t + 0 0 , 0 G

hence the realised model contains q 3( + ) 6h0 1 h0 √ h0+1 f (G) = G + q0G 4 + p0G . h0(1 + h0)

In this model, he “Big Rip” singularity occurs at some moment t = ts for any value of h0 6= 0, 1, 3; it is interesting that the “Big Rip” singularity does not occur if h0 = 1. 2 Type I singularity when β > 1. In this case, function p(t) is given as an approximation, 1 p(t) ' − 2 −2β , 4h0(ts − t) for h0 6= 0. Extracting the coordinate time as a function of the Gauss-Bonnet invariant from the constraint, we may write the realised model as r G f (G) = −12 . 24

Here, “Big Rip” may also occur, for any value of h0 6= 0. 3 Type II, III and IV singularities when β < 1. Here, solving the differential equations is achieved by approximation, 1 ( ) ' p t −1−β , 2h0(1 + β)(ts − t) √ 3 √ 1 h = 1 also yields analytic solutions, where f (G) = G ln(r G), where r some positive con- 0 2 0 0 stant. 84 the gauss-bonnet theories of gravity

for h0 6= 0 and β 6= −1. The realised model takes the form

2 2β 6h (3β + 1) |G| 3β+1 f (G) = 0 . + 3 β 1 24h0|β|

Following t → ts, a “Big Freeze” (type III) occurs, for large values of G - 3β + 1 2β 1 thus, in the large curvature limit- if > 0 and 0 < < , both β + 1 3β + 1 2 of them yielding 0 < β < 1. Another case could be the “Sudden” singularity (type II), appearing for 3β + 1 2β 1 > 0 and −∞ < < −1, which eventually leads to − < β < β + 1 3β + 1 3 3β + 1 2β 0; in a similar manner, if < 0 and −∞ < < −1, reading β + 1 3β + 1 1 −1 < β < − , the type II singularity will also emerge in t → t . 3 s 3β + 1 2 2β Finally, if > 0 and < < 1, then in the large curvature limit, β + 1 3 3β + 1 the type IV singularity appears as t → ts.

In general, all kinds of singularities appear in the R + f (G) cosmological mod- α els, especially is f (G) = f0|G| [289]. In that case, the results obtained in the case of minimal coupling about the phantom case and the possibilities of a “Big Rip”, are still accurate; it is, though, still possible to generate a deSitter expansion phase, even if the evolution of the Universe initiates with a deceleration corresponding 1 to radiation- or matter-dominated eras (for w > − ). 3 Part III

RECOSNSTRUCTIONOFCOSMOLOGICALMODELS

From the presented modiﬁed theories, the f (R) and the f (G) are considered. 1. A technique of reconstruction is applied on the f (R) theory, real- ising many cosmological models and accounting for their viability. 2. Given the ghost-free case of f (R, G) theory of gravity, three cosmological models are realised and their viability is proved.

THE f (R) THEORYUNDERTESTING 4

In this chapter, we arrange for a first attempt to prove the validity and viability of the f (R) theory in its simplest form. Our interest lies on the ability to reconstruct viable cosmological models, reflecting specific cosmological eras, from the scratch by means of the f (R) theory. In doing so, we shall employ a well known reconstruction technique, by [305], that allows to obtain the f (R) function in a closed form from the Friedmann equations and a phenomenological Hubble rate, given as a function of time -or of the e-foldings number. Initially, we shall present this technique and then move onto some examples, in order to account for its validity and to prove the usefulness of the respective theory of gravity.

4.1 the reconstruction technique

The reconstruction technique by [305] assumes the following. • The mass-energy density of the Universe can be given as a sum of mass- energy densities of several ﬂuids that are contained in the Universe, as

−3(1+wi) −3(1+wi) −3(1+wi)N ρ = ∑ ρi = ∑ ρi(0)a = ∑ ρi(0)a0 e ,(229) i i i

where ρi(0) the mass-energy density of the i-th fluid at some moment t0, a0 the scale factor at that moment, and wi the barotropic index of the i-th fluid. • We assume a flat FLRW background, hence the Hubble rate is uniform. It can be given as a unique, integrable and reversible function of time, H(t). Hence, we can always define the e-foldings number, N, and rewritte the Hubble rate as H(N) = h (− ln(1 + z)) .(230)

• Due to the ﬂat FLRW background, the Ricci scalar is given as a function of time, since R = 6H˙ + 12H2 .(231) As a result, the Ricci scalar is also expressed as a function of the e-foldings number, N. Furthermore, the Ricci scalar must be a reversible of time, and by extend of the e-foldings number, so that the latter can be written with respect to it, as N = N(R). • The f (R) theory holds. As a result, the Friedmann equation of the f (R) theory of gravity reads 1 184H2 H˙ + HH¨ f (R) − 3H2 + H˙ f (R) + f (R) = κρ ,(232) RR R 2

87 88 the f (r) theory under testing

∂ f ∂2 f where f (R) = and f (R) = . R ∂R RR ∂R2 The scope of the reconstruction technique is to solve the last equation with respect to f (R). In order to do so, the Hubble rate must be expressed with respect to the e-foldings number and the e-foldings number must be expressed with respect to the Ricci scalar. If all these occur, then the Friedmann equation turn a non- autonomous linear differential equation, that usually admits solutions by means of hypergeometric functions or special polynomials.

4.1.1 Inﬂationary dynamics of f (R) gravity: Formalism

In this section we present the formalism of slow-roll inﬂationary dynamics in the context of the f (R) gravity. For further details on the subject, the reader can review refs. [194, 196, 288], and also refs. [313, 315] for recent results. The slow-roll indices ei, i = 1, ..., 4 for a general vacuum slow-roll f (R) gravity are,

H˙ F˙ E˙ = − = = R = e1 2 , e2 0 , e3 , e4 ,(233) H 2HFR 2HE with the function E being equal to,

3F˙ 2 E = R .(234) 2κ2 The observational indices for such a model though, are not the slow-roll indices, but two observable quantities related to them, the spectral index and the scalar- to-tensor ratio. Their calculation may vary, depending on the values the slow-roll indices acquire during the slow-roll era. In the general case, considering that the slow-roll indices are naturally small (so small as to satisfy e˙i ' 0), the spectral index of the primordial curvature perturbation is given as [194, 196, 288],

ns = 4 − 2νs ,(235)

with the quantity νs being equal to, s 1 (1 + e − e + e )(2 − e + e ) = + 1 3 4 3 4 νs 2 .(236) 4 (1 − e1)

In a particular case, where ei 1, the spectral index is approximately equal to,

ns ' 1 − 4e1 + 2e3 − 2e4 ,(237)

which will not be our case, if we are to preserve generality. In addition, the scalar- to-tensor ratio r for a vacuum F(R) gravity is deﬁned as follows [303],

8κ2Q r = s ,(238) FR 4.1 the reconstruction technique 89

where Qs is a quantity related to the function E as E = Qs 2 2 .(239) FR H (1 + e3) It is easy to show that, in cases such as this, the scalar-to-tensor ratio reads,

48e2 = 3 r 2 .(240) (1 + e3)

In the particular case that ei 1, the scalar-to-tensor is simpliﬁed even more, since e1 ' −e3 and the above relation becomes,

2 r = 48e1 .(241)

Before proceeding to our analysis of the F(R) reconstruction, we should recall the observational constraints on the spectral index ns and the scalar-to-tensor ratio r, that have been acquired from the Planck data [13], being

ns = 0.9649 ± 0.0042 , r < 0.063 (242) at 68 % conﬁdence level, while the BICEP2/Keck-Array data [7] constrain the scalar-to-tensor ratio as follows,

r < 0.07 , (243) at 95% conﬁdence level. Also, from eq. (242) it is obvious that the spectral index can be considered as compatible with the Planck observations, when it takes values in the interval ns = [0.9595, 0.9693].

In the rest of this section we shall investigate the behavior of the slow-roll indices during the slow-roll era. Although, it is appropriate to use the deﬁnition of the observational indices as described above, the viability of the theory is independent of the choice of the condition ei 1, i = 1, ..., 4 the slow-roll indices should satisfy, no matter of the approximation chosen with regard to the observational indices, and namely eqs. (235) and (237) for the spectral index, or eqs. (238) and (240) for the scalar-to-tensor ratio. In order to be as accurate as possible, we choose the formally more rigid approach, thus the spectral index is given by eq. (235) and the scalar-to-tensor ratio is given by eq. (240). Secondly, a further simpliﬁcations of the slow-roll indices appearing in eq. (233) is possivle ; after some algebra we obtain,

H˙ F˙ F R¨ = − = = RR ˙ + ¨ = RRR ˙ + e1 2 , e2 0 , e3 24HH H , e4 R ,(244) H 2HFR HFR HR˙ ∂2F ∂3F with F = and F = . Since our analysis focuses on early-time accel- RR ∂R2 RRR ∂R3 eration, the e-foldings number N is a much more convenient variable that time 90 the f (r) theory under testing

t, thus we may express the above quantities in terms of N, using the following differentiation rules, d d = H .(245) dt dN d2 d2 dH d = H2 + H ,(246) dt2 dN2 dN dN By their use, the slow-roll indices become,

H0(N) e = − , e = 0 , (247) 1 H(N) 2 00 2 0 0 2 2 FRRR 6H(N)H (N) + 24H(N) H (N) + 6H(N)H (N) e3 = , FR 2H(N) 0 2 0 2 −H00(N) + 2H (N) − H (N) H(N) H(N) FRRR 00 0 2 0 e4 = − 3e1 + 6H(N)H (N) + 6H (N) + 24H(N)H (N) . H(N)e1 FR The form of the these equations is such that knowing the Hubble rate as a function of the e-foldings number, H(N), as well as the F(R) gravity generating the evolution H(N), one can easily calculate the slow-roll indices and the corresponding observational indices, by merely substituting those in the above equations. In order to proceed to the aforementioned calculations, we need to express the Hubble rate as a function of the e-foldings number N ; this is easily found by solving the equation N = ln a with respect to the cosmic time and substituting the result in each part of the given Hubble rate. Furthermore, in order to describe the evolution of space-time and use it to conclude to the slow-roll indices and the observational indices, we shall need the functional form of the F(R) gravity. To acquire this, a reconstruction technique has been proposed by [305], by which we shall also proceed. The cosmological equation (108), can take the following form,

F(R) − 18 4H(t)2 H˙ (t) + H(t)H¨ (t) F (R) + 3 H2(t) + H˙ (t) F (R) − = 0 , RR R 2 (248) The equation is transformed, using the e-foldings number N, and also the differentiation rules of Eqs. (245) and (246), the eq. (108), as follows,

3 0 2 0 2 3 00 − 18 4H (N)H (N) + H (N)(H ) + H (N)H (N) FRR(R) (249) F(R) + 3 H2(N) + H(N)H0(N) F (R) − = 0 , R 2 where the primes stand for H0 = dH/dN and H00 = d2 H/dN2. Furthermore, introducing the function G(N) = H2(N), we may write the differential equation (249) in terms of G(N), thus obtaining,

0 00 − 9G(N(R)) 4G (N(R)) + G (N(R)) FRR(R)+ (250) 3 F(R) + 3G(N) + G0(N(R)) F (R) − = 0 , (251) 2 R 2 4.2 the desitter and the quasi-desitter expansion 91 where G0(N) = dG(N)/dN and G00(N) = d2G(N)/dN2. The Ricci scalar R can also be expressed as a function of G(N) as,

R = 3G0(N) + 12G(N) .(252)

Hence, the F(R) gravity which realizes the Hubble rate H(N) can be found by solving the differential equation (250). Accordingly, we can ﬁnd the quantity FRRR FR in terms of the Ricci scalar ; expressing the Ricci scalar as a function of N by using eq. (252), we can calculate the exact form of the slow-roll indices (247), thus the observational indices are easily obtained.

4.2 the desitter and the quasi-desitter expansion

The deSitter expansion is one of the oldest cosmological solutions and concerns the constant rapid expansion of an empty 3-d space. In terms of the ﬂat FLRW Universe, this can be represented by a constant Hubble rate,

H = H0 (253)

Such a cosmological model is considered to govern the late-time accelerating phase of the Universe, also named deSitter phase due to the obvious similarity. We shall attempt to realize a cosmological model fro an empty ﬂat FLRW space- time under this Hubble rate and the f (R) theory. First, we should notice that this Hubble rate is not a function of time, hence no reason exists for its rewriting with respect to the e-foldings number. Hence, its time derivatives are simply H˙ = H¨ = 0, and substituting them to 248, we obtain the very simple differential equation

2 6H0 fR(R) − f (R) = 0 , (254) whose solution is simply R 6H2 f (R) = C1e 0 ,(255) with C1 being an integration constant. Choosing carefully C1 = 1 and assuming H0 ' 70 (the current value), we easily obtain an f (R) function very close to f (R) = R, a result we merely expected, since General Relativity can itself explain very well the current deSitter phase, if one assumes the existence of the Cosmological constant -inherited here via the current value of the Hubble rate. Furthermore, the effective barotropic index turns we f f = −1 at any case, another anticipated result, since the assumption of a Comsological constant inherent to the deSitter Hubble rate goes along with negative pressure, equal in magnitude to its energy density.

Having successfully realised the late-time accelerating expansion via the constant Hubble rate, we will assume a rather more intriguing case, with a linear 92 the f (r) theory under testing

Hubble rate, that may interpret the early-time accelerating expansion, namely the inﬂation. Here, we will also address the issue of reconstructing the slow-roll parameters. The linear Hubble rate is given as

H(t) = H0 − H1t .(256)

Integrating with respect to cosmic time, we obtain the e-foldings number, and reversing it we may have  q − + 2 −  H0 2C0 H1 H0 2H1N   H t = q 1 ,(257)  H + 2C H + H2 − 2H N  0 0 1 0 1  H1

where C0 the integration constant; of the two we choose the latter, since cosmic time must increase with the increase of e-folds. Substituting this to eq. (256), we obtain q 2 H(N) = 2C0 H1 + H0 − 2H1N .(258) As a result, the Ricci scalar is given as

2 R(N) = 6 H1(4C0 − 4N − 1) + 2H0 .(259)

Reversing the latter, we eventually have the e-foldings number as a function of the Ricci scalar 24C H + 12H2 − 6H − R N = 0 1 0 1 .(260) 24H1 Substituting to the Hubble rate, and then utilizing eq. (249), we obtain the differential equation

24H1(R + 6H1) fRR(R) + (R − 6H1) fR(R) − 2 f (R) = 0 , (261)

that can be solved analytically only by means of inverse functions. However, since our focus lies with early stages of the Universe and thus with large curvature, we may reduce our differential equation to

R fR(R) − 2 f (R) = 0 . (262)

Of course, the latter’s solution is easily extracted as

2 f (R) = C1R ,(263)

where C1 is the integration constant. This result, again, is clearly anticipated, since it corresponds to the Starobinsky quadratic term that is used to generate inﬂation and move away the singularity at the initiation of cosmic time; as we stated many times, such a term is expected to dominate the early phase of the Universe. The 4.3 the radiation-dominated and the matter-dominated eras 93

1 = integration constant is hence equal to C1 2 , hence it is expected to be 6MPlanck very large. Using the slow-roll parameters deﬁned in eqs. (247), we may write the spectral index and the tensor-to-scalar ratio for the quasi-deSitter expansion from eqs. (237) and (240) respectively; they are easily found to be v u 22 u H1(6C0 − 6N − 1) + 3H0 ns = 4 − t ,(264) 22 H1(2C0 − 2N − 1) + H0 r = 0 . (265)

The tensor-to-scalar ratio is conﬁned to zero, hence it always agrees with the constraints of Planck 2018 and BICEP2/Keck-Array. As for the spectral index, it is usually very close to 1, since the square root is approximately equal to 3; there are very few values of C0 for N > 50 and given H0 and H1, that match the Plack 2018 data, so very careful ﬁne-tuning is required. It is worth mentioning that, as in Starobinsky’s model, the C1 parameter -originally the coupling constant in the Lagrangian- plays no role in the determination of the observables.

4.3 the radiation-dominated and the matter-dominated eras

Let us now turn our focus on the intermediate eras, those very well explained in terms of relativistic FLRW cosmologies, assuming a simple ideal fluid that dominates the Universe. We know that the Universe contains two types of fluid, 1 a hot relativistic one (radiation) with w = (radiation), and a cold pressureless 3 one with w = 0 (matter); it may also contain small amounts of Bose-Einstein condensates, that are considered as stiff matter, with w = 1. All of the above are well known to follow a specific decrease in their mass-energy density, as depicted here,

−3 −3N ρm(N) = ρm(0)a0 e , −4 −4N ρr(N) = ρr(0)a0 e and −6 −6N ρs(N) = ρs(0)a0 e , where ρi(0) the mass-energy density of the specific fluid and a0 the scale factor at some specific moment of cosmic time, e.g. today -and m, r and s indices going for matter, radiation and stiff matter respectively. From these and the relativistic Friedmann equation, eq. (19), we may construct a Hubble rate of the form

r κ H(N) = H2 + ρ a−3e−3N + ρ a−4e−4N + ρ a−6e−6N ,(266) 0 3 m(0) 0 r(0) 0 s(0) 0 94 the f (r) theory under testing

where H0 the current value of the Hubble rate. The above Hubble rate is a function of the e-foldings number already, consequently the Ricci scalar will immediately be −6N 3 3N κe a0e ρm(0) − 2ρs(0) ( ) = 2 + R N 12H0 6 .(267) a0 Reversing it, we may obtain  √ q  a3 κ 96H2ρ + κρ2 − 8ρ R − a3κρ 1 0 0 s(0) m(0) s(0) 0 m(0) = − N ln  6 2 6  .(268) 3 2 12a0 H0 − a0R

Substituting this to the Hubble rate and utilizing the Friedmann equation, eq. (249), we obtain

3 6 2 3 2 6κ(a0ρm(0) − 4ρs(0)) 3a0 H0 + a0κρm(0) + a0κρr(0) + κρs(0) fRR(R)+ 1 2 9 4 (269) + 6a02H0 − a0κρm(0) − 2a0κρr(0) + 4a0κρs(0) fR(R)− 1 − a02 f (R) = 0 , which is a linear autonomous differential equation; its solution is easily extracted by means of exponentials, so

q+ R q− R f (R) = C1e + C2e ,(270)

where 12 2 9 8 6 −6a0 H0 + a0κρm(0) + 2a0κρr(0) + 4a0κρs(0) q± = ± 3 6 2 3 2 12κ a0ρm(0) − 4ρs(0) 3a0 H0 + a0κρm(0) + a0κρr(0) + κρs(0) q 12 ± a0 Σ .

and 1 Σ = × 3 6 2 3 2 12κ a0ρm(0) − 4ρs(0) 3a0 H0 + a0κρm(0) + a0κρr(0) + κρs(0) 12 4 9 2 8 2 6 2 2 36a0 H0 + 60a0 H0 κρm(0) − 24a0 H0 κρr(0) + a0κ 25κρm(0) − 336H0 ρs(0) + 5 2 4 2 2 3 2 2 2 2 2 + 28a0κ ρm(0)ρr(0) + 4a0κ ρr(0) − 64a0κ ρm(0)ρs(0) − 80a0κ ρr(0)ρs(0) − 80κ ρs(0)

It is easy to formulate these parameters, by appropriately choosing C1 and C2, as well as the observational values for H0, a0, ρm(0), ρr(0) and ρs(0), so that the behaviour of f (R) will be identical to the curvature generated by relativistic models containing these ideal fluids. We can see that, in the context of f (R) gravity, even with absence of actual matter fields the respective cosmological models can be realised, due to the modified curvature terms. 4.4 an exponential early-time expansion 95

4.4 an exponential early-time expansion

The quintessential inﬂation scenario is quite popular in the literature [48, 71, 98, 113, 168, 169, 202, 205, 208, 229, 230, 294, 306–308, 311, 411], especially in the context of scalar-tensor theory. In this paper, we shall be interested in the quintessential inﬂation scenario developed in refs. [176, 178, 179], according to which, the Hubble rate is equal to,

 (− 2 + ) 3HE Λ t  H HEe E t ≤ 0  √ √ r H(t) = Λ 3HE + 3Λ tanh 3Λt ,(271)  √ √ t ≥ 0  3  3HE tanh 3Λt + 3Λ while the corresponding scale factor is,

 (−3H2 +Λ)t H2 h E −1i  E e HE  − H2 + a e 3 E Λ t ≤ 0 a(t) = E ,(272) √ √ 1  3HE 3 aE √ sinh 3Λt + cosh 3Λt t ≥ 0  3Λ where HE, aE and Λ are real and positive constants of the theory. The model of eq. (271) describes two eras, the “negative time” era where the Hubble rate follows an exponential evolution, and the “positive time” era where the Hubble rate turns is a function of hyperbolic tangents. As we stated in previous chapters, for negative times, the effective equation of state (EoS) parameter we f f which is deﬁned as,

2H˙ w = −1 − (273) e f f 3H2 and turns equal to,

2 2 (−3HE+Λ)t 2(−3HE + Λ)e we f f = −1 + .(274) 3HE In this section we shall investigate how the early-time part of this scenario can be realized in the context of f (R) gravity, that corresponds to the “negative-time” era, where the Hubble rate is an exponential; in the following one, we shall deal with the “positive-time” part of the Hubble rate, that may describe the late-time evolution of the Universe. Solving eq. N = ln a and substituting the result to eq. (271), the resulting expression for the Hubble rate is,   Λ  − 3HE + H (N − C0) t ≤ 0  E√  √ ( + ) ( + ) √  H2 e6 C0 N − e6 C0 N +e12N + H e6C0  3Λ 3 E Λ 3Λ E ( ) = 3HE± H N q e6C0 H2 +e6N ,(275) Λ 3 E √  ( + ) ( + ) √ t ≥ 0  3 H H2 e6 C0 N − e6 C0 N +e12N + H e6C0  √ 3 E 3 E Λ 3Λ E  3Λ±  e6C0 H2 +e6N 3 E 96 the f (r) theory under testing

where C0 stands for the integrating constant in the definition of the e-foldings number. We observe that the Hubble rate in respect to the e-foldings number N has not only two but three distinct cases, one for “negative time” and tow for“positive”. This unorthodox result should be attributed to the unnatural form of the Hubble rate in respect to time, for “positive time” - since the hyperbolic tangents appearing when t ≥ 0 are not inversed by means of smooth functions. This forces us into splitting our analysis naturally for “negative” and “positive times”, so that the differences of the two types of inflation are made obvious ; each of them leads to a both similar and different form of the F(R) gravity that realizes the corresponding inflation. For the inflationary era, the relevant part of the Hubble rate as a function of the e-foldings number N is the one corresponding to the “negative times”, which is, Λ H(N) = − 3HE + (N − C0) , HE from which we observe that the Hubble rate is a linear function of N. From Eq. (252), we may easily acquire the Ricci scalar in the case of “negative times”, which results to depend on the square of the total number of e-foldings, and precisely,

Λ 2 R(N) = 6 − 3HE + (1 + 2N − 2C0)(N − C0) .(276) HE Solving this for the number of e-foldings, N, we may express the latter as a function of the curvature scalar, so as to use it for the Eq. (??). Due to the square of the curvature scalar, we get two solutions, notably, v 1 1 u 46 N (R) = C + ± u R − 36 . (277) 1,2 0 4 24t 2 −3H + Λ E HE

The above two solutions are actually presenting us with two completely different cases: on the one hand, the e-foldings number increases with the curvature, while on the other it decreases.

1.00 0.10

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0.95 0.00 0 20 40 60 80 100 0 20 40 60 80 100

C0 C0

Figure 4: The spectral index and the scalar-to-tensor ratio for “negative times” with respect to C0 for N = 50 and C1 = C2 = 1. 4.4 an exponential early-time expansion 97

Since, both of them cannot be accurate, the second one must be simply objected. Yet, its non-physical meaning might as well arise naturally, since no inﬂationary scenario can be achieved via the reconstruction technique, if this case is adopted. The cosmological equation yielding the functional form of the f (R) gravity for the case of “negative times”, is easily acquired if we substitute Eq. (276) and (277) in Eq. (250). Its initial form is rather obscure and hard to handle. However, due to the fact that the “negative times” case describes the inﬂationary era, we may focus on the large curvature limit. For the increasing number of e-foldings, we obtain the following form of the differential equation (250),

R2 1 f (R) − f (R) = 0 , (278) 2 RR 2 which is a simple case of the second-order Euler-Cauchy differential equation. Its solution is easy to admit in terms of a power law, and it naturally has the form,

√ √ 1+ 5 1− 5 f−(R) = C1R 2 + C2R 2 .(279)

This solution is very close to F(R) ∼ R2, which is a good approximation for the inflationary era. If we consider the case of decreasing number of e-foldings, aside its non-physical meaning, we may end to a differential equation of the following form R2 R 1 f (R) + f (R) + F(R) = 0 , (280) 2 RR 2ε R 2 where ε is vary small and naturally going to zero. As a result, this differential equation presents a singularity, since the first-order term goes to infinity at all cases. Its solution may be given in the form of a power-law as before, since it also comes in the form of a second-order Euler-Cauchy differential equation,

(− 2 + )2 5(−3H2 +Λ)2 4 3HE Λ E 2 f−(R) = C1R ε + C2R ε , which is led to inﬁnity for all cases - since ε → 0. This case presents no mathematical interest, as well as no physical meaning, so we will focus on the solution of Eq (279). Continuing on with the viable solution of Eq. (279), we can turn our concern to the slow-roll indices. Given the Hubble rate for “negative times”, we come up with the following form of the slow-roll indices,

1 e1 = , e2 = 0 , (281) C0 − N 3 ∂ f− 2 4 ∂R3 −3HE + Λ 2 e3 = 9 (−4C0 + 4N + 1) , ∂ f− H4 ∂R E 3 ∂ f− 2 2 3 −3H + Λ 2 = ∂R E (− + + ) + e4 6 ∂ f 2 4C0 4N 1 − H C0 − N ∂R E 98 the f (r) theory under testing

Substituting the functional form of the F(R) gravity realized by the Hubble rate of Eq. (271) for “negative times”, and using Eq. (235) and (238), we arrive to the spectral index and the tensor-to-scalar ratio of our inﬂationary model, as functions of the number of e-foldings, N, and ﬁve parameters, HE and Λ from the Hubble rate and the integration constants C0, C1 and C2. However we do not quote the result since it is quite lengthy and complicated to quote it here.

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0.99 0.08

0.98 0.06 s n

0.97 0.04

0.96 0.02

0.95 0.00 0 20 40 60 80 100 0 20 40 60 80 100

C0 C0

Figure 5: The spectral index and the scalar-to-tensor ratio for “negative times” with respect to C0 for N = 60 and C1 = C2 = 1.

In order to investigate the viability of the model and whether it can actually describe the inflationary era, we need to compare the spectral index and the tensor- to-scalar ratio derived from it with the observation data coming from Planck and BICEP2/Keck-Array collaborations and prove that there are values of our five parameters for which both indices coincide with the values appointed unto them by the observations. Such an investigation is in fact easier, considering that integration constants from solving the differential equation do not really play an important role, so by setting C1 = C2 = 1 makes things a lot easier. Furthermore, setting ns (theoretical) = 0.9644 for fixed values of the parameters HE and Λ and upon solving for C0, hands out the result  47.0884 N = 50 C0 = 57.0884 N = 60 ,

which restraints signiﬁcantly the parameter space we need to check in order to prove the viability of our model 1.

The general behavior of the spectral index and the tensor-to-scalar ratio over the C0 parameter can be seen in Figs. 4 and 5, where we plot the behavior of the spectral index and of the scalar-to-tensor ratio for N = 50 and N = 60 respectively, for C1 = C2 = 1 and for various values of HE and Λ satisfying the 2 constraint 3HE < Λ. We need to notice that in Figs. 4 and 5, the solid red line

1 The result was derived numerically, so it is not the only solution of the equation. In fact, we know and we can see from the following plots that three solutions exist. We merely focus on this one obtained by our numerical analysis, without rejecting the viability of the others. 4.4 an exponential early-time expansion 99

stands for the observable ns = 0.9644, while the dashed red lines cover the con- ﬁdence interval [0.9595, 0.9693]. As for the black line, it corresponds to r = 0.07, which is the top constraint from the BICEP2/Keck-Array observations.

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0.98 0.98 s s n n

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0.95 0.95 -15 -10 -5 0 5 10 15 0 200 400 600 800 1000

He Ë

Figure 6: The spectral index for “negative times” with respect to HE in the left plot, and to Λ in the right plot, for N = 50, C0 = 47 and C1 = C2 = 1.

What we perceive from these graphs is that while the tensor-to-scalar ratio of our model is mostly below the observable constraint -surpassing it only when C0 ' N-, the spectral index is compatible to the observations for only a small interval of values of C0, around this very point. As a result, we should focus to this small interval -using our proposed values, C0 = 47 for N = 50 and C0 = 57 for N = 60, and investigate whether the model is indeed viable in there. Setting N = 50, C0 = 47 and C1 = C2 = 1, we may derive the plots appearing in ﬁgs. 6 of the spectral index with respect to the parameters HE and Λ of the Hubble rate, which represent the functional behavior of the spectral index with respect to HE (left plot) , and with respect to Λ (right plot), for N = 50, C0 = 47 and C1 = C2 = 1. Also in ﬁg. 7, we present the functional behavior of the tensor-to- scalar ratio as a function of HE (left plot) , and as a function of Λ (right plot), for N = 50, C0 = 47 and C1 = C2 = 1.

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He Ë

Figure 7: The tensor-to-scalar ratio for “negative times” with respect to HE in the left, and to Λ in the right, for N = 50, C0 = 47 and C1 = C2 = 1.

Again, in ﬁgs. 6 and 7 the solid red line stands for the observable ns = 0.9644, while the dashed red lines denotes the conﬁdence interval [0.9595, 0.9693], and the black line corresponds to r = 0.07. It is interesting to observe that, for such a value 100 the f (r) theory under testing

of C0, the majority of HE and Λ values realize a viable inflationary model, within the limits of the observational constraints. In fact, the only cases that disagree 2 with the observations are the extreme cases where 3HE = Λ, when the Hubble rate becomes constant over time and two of the slow-roll indices become zero, turning the observational indices to infinity. All other cases, whether the Hubble 2 2 rate is increasing (3HE < Λ) or decreasing (3HE > Λ) with time, are viable. Finally, it is compelling to have simultaneous viability of the observational indices for the same values of the free parameters, namely for the same values of C0, HE and Λ. This can be easily shown with a number of parametric plots, where the tensor-to-scalar ratio is plotted against the spectral index with one parameter varying while the rest are fixed. Once more, we will consider N = 50 and N = 60 separately and fix C1 = C2 = 1, so in fig. 8 we present the parametric plots of the spectral index (horizontal axis) and of the tensor-to-scalar ratio (vertical axis), for HE = 20 and Λ varies in the range of values (0, 1300), and in fig. 9 the case that Λ = 640 and HE varies in range (0, 100) is considered. As in previous plots, in figs. 8 and 9, the solid and dashed red lines depict the values of ns derived from observations, while the black line stands for the top limit set by observations to r - the dashed black line denotes r = 0.1. Indeed, we conclude that the coincidence of the viability for both observable indices for a large range of values of HE and Λ, occurs when C0 is close to 47.0883 for N = 50, or to 57.0883 for N = 60. Hence, the free parameter C0 must take values close to the value of the e-foldings number N. Interestingly enough, as a further hint to this viability, the spectral index remains higher than 0.93 for all values of the three parameters considered, and tends to 1, only when the tensor-to-scalar ratio tends to zero. In conclusion, we demonstrated that a specific class of quintessential inflation model, can be compatible with the observational data, for a wide range of parameters, without assuming that the slow-roll condition holds true. This result indicates that there exist several theoretical proposals of Hubble rates that can provide a viable early-time phenomenology in the context of several different theoretical frameworks.

4.5 a hyperbolic tangent late-time evolution

Now, we shall turn our concerns towards the “positive time” evolution, described by the Hubble rate

√ √ √ 2 6(C +N) 6(C +N) 12N 6C 3Λ 3H e 0 −Λe 0 +e + 3ΛHEe 0 r H ± E 3 E 6C0 2 6N Λ 3e HE+e H(N) = √ √ . 3 √ 3H 3H2 e6(C0+N)−Λe6(C0+N)+e12N + 3ΛH e6C0 ± E E E 3Λ 6C0 2 6N 3e HE+e This part of the Hubble rate follows a rapid exponential evolution with incresing number of e-foldings. Cases such as this have not been considered in familiar literature, mainly due to their complexity; however, such a Hubble rate could 4.5 a hyperbolic tangent late-time evolution 101 possibly explain the classical cosmic eras and -mainly- the late-time accelerating expansion of the Universe. As in the case of “negative times”, we begin from Eq. (252), acquiring the Ricci scalar in the following form √ 6N 6C0 2 6N 6C0 3 6N 6Λe e Λ − 3HE + 2e e 9HE − 3HEΛ − 3ΛRS + 3HEe R(N) = r √ 3 6N 6N e RS 3HERS ± 3Λe √ 6C0 4 2 6N 2 e 9HE − 3HEΛ + e 3HE + Λ ± 2 3ΛHERS , √ 3 6N RS 3HERS ± 3Λe (282) where q 6N 6C0 2 6N RS = e e 3HE − Λ + e If we are to solve this for the number of e-foldings, N, so as to express the latter as a function of the curvature scalar, we will end up with the following four foms   q  6 (− H2 + )  2Λ 3 E Λ C0 ln ± √  N(R) = 6 R − 4Λ (283)  q  6 2 C0 ln ± Λ − 3HE .

The choice of the sign should actually follow the relation between the parameters, R H and Λ ; naturally, we would expect that Λ > 3H2 and Λ < , so that the sixth- E E 4 roots could be naturally deﬁned and the number of e-foldings turns a positive real number. For the above four solutions, we can restrict our research to two of them, those bearing the positive sign ; interestingly, the other two give the very same F(R) realization, so it does not make any difference. From these two, only the q 6 2 case where N(R) = C0 ln Λ − 3HE seems to present some physical meaning, though eventually a constant number of e-foldings has no meaning and realizes 1 a F(R) = 0 model. The other case, where N(R) ∼ ln(R 6 ), despite its “unnatural”, given a careful choice of parameters, it comes to realize -as we see afterwards- an interesting form of F(R) gravity. Taking this last case, we can easily extract the functional form of the f (R) gravity for the case of “positive time”, substituting Eq. (276) and (277) in Eq. (250) and acquiring the differential equation of the cosmological evolution. Consider- ing once more the case of slow-rolling, we focus on the large curvature limit and we obtain the following form

2 6R fRR(R) − 2R fR(R) + F(R) = 0 , (284) 102 the f (r) theory under testing

which is nothing but a second-order Euler-Cauchy differential equation, whose solution is extracted in terms of a power law, as follows

√ √ 2+ 10 2− 10 fpositive(R) = C1R 3 + C2R 3 .(285)

This solution is close to f (R) ∼ R2 + R−1, which contains both a Starobinsky term (the squared part) that may generate early-time dynamics, but also an unnatural solution for late-time evolution (the inverse part). Interestingly enough, if we think out of the inﬂationary era and do not apply the slow-roll condition, the original differential equation of the cosmological evolution can be still solved analytically and its solution is given in terms of hypergeometric functions √ √ ! 4 + 10 4 − 10 1 R f (R) =C F − , − ; ; − 2 poritive 1 2 1 6 6 6 2Λ √ √ ! (286) R 5/6 1 − 10 1 + 10 11 R + 2 − C F , ; ; − 2 2Λ 2 2 1 6 6 6 2Λ

Such a solution, although uncomparable with the Planck and BICEP2/Keck-Array data, and probably uncompatible with the inﬂationary era, is still very important. According to ref. [305], f (R) gravity models realizing the radiation-dominated era, the matter-dominated era or the late-time evolution are likely to be given in closed form as hypergeometric functions. Thus, we indeed were right to understand this part of the Hubble rate (271) for “positive times”, as the part of the Hubble rate realizing the cosmological eras after inﬂation, even that of the late-time accelerating expansion.

4.6 conclusions

In this chapter, we investigated the reconstruction of f (R) models, based on the well-known technique due to [305], allowing us to derive the f (R) gravity in closed form for a number of given Hubble rates. Initially, we considered Hub- ble rates taken from classical cosmology, such as the deSitter expansion constant Hubble rate, the quasi-deSitter linear Hubble rate, or the radiation-dominated and matter-dominated eras decreasing Hubble rates. Reconstructing the f (R) theory, we arrived to the result that it mimics the results obtained from classical and relativistic cosmology in the context of a flat FLRW space-time, even if no matter fields are present. The four major cosmic eras of the Standard Cosmo- logical model (inflation, radiation-domination, matter-domination, dark-energy- domination) are retrieved and classification is made possible. In the following, we asume the Hubble rate to be given by a piecewise complex form due to [176, 178, 179], the “negative time” part representing early-time evolution of the Universe, and the “positive time” part representing late-time. 4.6 conclusions 103

This specific form of the Hubble rate is arrived from a quintessntial model of inflation and accelerating expansion. We demonstrated how both quintessential models can be realized in the context of vacuum f (R) gravity. The quintessential inflation model was rigorously tested for its early-time viability by confronting the resulting theory with the observational data; particularly, we calculated the slow-roll indices by using the e-foldings number as the main variable, and also we calculated the spectral index of the primordial curvature perturbations as well as the tensor-to-scalar ratio. As we showed, there is a large range of parameters values for which the resulting inflationary theory is compatible with both the latest Planck and BICEP2/Keck-Array data, with one single exception - the C0 integration constant needs to be close to the e-foldings number if the viability is to coincide for both indices. The late-time viabilit of the quintessential model was also tested by comparing it to the canonical deSitter evoltuion. 104 the f (r) theory under testing

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Figure 8: The tensor-to-scalar ratio against the spectral index for N = 50 in the left, and N = 60 in the right, for C1 = C2 = 1, HE = 20 and varying C0, in the range C0 = [44, 49] in the left, and C0 = [54, 59] in the right ; Λ takes respectably the values 20, 80, 320 and 1280, but the four separate curves coincide. 4.6 conclusions 105

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0.02

0.00 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Spectral Index

Figure 9: The tensor-to-scalar ratio against the spectral index for N = 50 in the left, and N = 60 in the right, for C1 = C2 = 1, Λ = 640 and varying C0, in the range C0 = [44, 49] in the left, and C0 = [54, 59] in the right plot. The parameter HE takes respectably the values 1, 9, 18 and 81, but the four separate curves coincide.

THEGHOST-FREEEINSTEIN-GAUSS-BONNETTHEORY 5

This chapter focuses on an inflationary model derived from a Ghost-free Gauss- Bonnet theory of Gravity, as analysed in [290] and chapter 3; that is a rather simple form of the scalar-Einstein-Gauss-Bonnet theories. Having shown already how the ghost modes can be effectively traced and eliminated by means of the Lagrange multipliers technique and specific constraints, we proceed here by presenting the inflationary dynamics of this theory for the general case (based on ref. [197]) and transforming them with respect to the e-foldings number, N, following a well-known reconstruction technique (see refs. [305, 348, 351], mainly for the f (R) case, which is mirrored here in the respective f (G) models). Us- ing the transformed slow-roll parameters, we calculate the spectral indices and the tensor-to-scalar ratio of the inflationary model and we are able to compare them to the Planck and BICEP2/Keck-Array data. The results that the viability of such an inflationary model depends on the choice of the coupling of the scalar field χ to the Gauss-Bonnet invariant G, in the form of a function h(χ). Choos- ing two simple functions -an exponential and a power-law- containing two free parameters, we prove that the inflationary model derived from the Ghost-free Gauss-Bonnet theory is viable for a constant (DeSitter), a linear (quasi-DeSitter) and an exponential (due to [176, 178, 179]) Hubble expansion rate. These results have been part of ref. [310].

5.1 inflationary dynamics

Given the scalar Einstein-Gauss-Bonnet action along with the constraint (eq. 207 or eq. 221), we can reconstruct the slow-roll dynamics of the inlfationary era for such a model and compare our results with the observations of Planck and BICEP/Keck-Array. In order to do so, we simply need to deﬁne the necessary quantities to match the observables, namely the spectral indices and the tensor-to- scalar ratio. In doing so, we follow [197] who explicitly present the perturbation theory for -inter alia- the scalar Einstein-Gauss-Bonnet theory with string and tachyonic corrections. Their analysis contains multiple couplings of the scalar ﬁeld, χ, with the curvature features, one of which is the coupling with the Gauss- Bonnet invariant ; excluding the rest, we maintain their results, adapting them in our case (see their Lagrangian, eq. (92), p. 12 in [197], and ours for comparison).

107 108 the ghost-free einstein-gauss-bonnet theory

We begin by deﬁning the functions Qi(χ) (eq. (103), p. 14 and eq. (106), p. 15), as 2 2 Qa(χ) = −4c1h˙ (χ)H = 8h˙ (χ)H , Qb(χ) = −8c1h˙ (χ)H = 16h˙ (χ)H ,(287)

Qc(χ) = 0 , Qd(χ) = 0 , Qe(χ) = −16c1h˙ (χ)H˙ = 32h˙ (χ)H˙ , Q f (χ) = 8c1 h¨(χ) − h˙ (χ)H = −16 h¨(χ) − h˙ (χ)H and 1 Q (χ) = F (R, χ) + Q (χ) = 1 + 8h˙ (χ)H . t 1 2 b

Afterwards, we may define the wave speeds cA (eq. (91), p. 11) and cT (eq (109), p. 15), as 2 ∂ f1 3F˙1 X + Q f c2 = ∂X 2F1 and c2 = 1 − , A 2 ˙ 2 T ∂ f1 2 ∂ f1 3F1 2F1 + Qb X + 2X 2 + ∂X ∂X 2F1 1 ∂ f λ ∂2 f where X = − χ˙ 2 , 1 = and 1 = 0. As a result, the two wave speeds 2 ∂X 2 ∂X2 are further simplified; the wave speed of the perturbed field is trivial as in the classical case, due to the simple form of the f (R, χ), 2 cA = 1 , (288) while the wave speed of the gravitational waves in non-trivial, 16h¨(χ) − h˙ (χ)H c2 = 1 + (289) T 2 + 16h˙ (χ)H In order to define the slow-roll parameters, we first need to define the function E(R, χ, X) as

2 2 4 F (R, χ) F˙1(R, χ) + Qa 192h˙ (χ) H E(R, χ, X) = 1 ω(χ)χ˙ 2 + 3 = −λχ˙ + . χ˙ 2F1(R, χ) + Qb 2χ˙ + 16χ˙h˙ (χ)H (290) The slow-roll parameters are deﬁned (using eq. (163-164), p. 20 of [197]), so that H˙ χ¨ e = , e = = 0 , (291) 1 H2 2 Hχ˙

1 F˙1(R, χ) 1 E˙ (R, χ, X) e3 = = 0 , e4 = , 2 HF1(R, χ) 2 HE(R, χ, X) 2 F˙1 + Qa 4h˙ (χ)H e5 = = and H 2F1(R, χ) + Qb H 1 + 8h˙ (χ)H

Q˙ t 4h¨(χ)H + 4h˙ (χ)H˙ e6 = = . 2HQt H 1 + 8h˙ (χ)H The two spectral indices, for scalar and for tensor perturbations in the inﬂationary era respectively, are deﬁned using the slow-roll parameters (see eq. (173-174), p. 21 in [197]), as

e1 − e2 + e3 − e4 e1 − e6 nS = 1 + 2 and nT = 2 .(292) 1 + e1 1 + e1 5.1 inflationary dynamics 109

Finally, we deﬁne the tensor-to-scalar ratio (eq. (179), p. 22 in [197]),

1 1 1 1 c 3 = − − ( + ) − + A r 4 e1 e3 2 2Qc Qd Qe Q f Q . 4F1(R, χ) H H 1 + b cT 2F1(R,χ) (293)

The above expressions of the parameters for the slow-roll inflationary dynamics, are in fact functions of time, t. However, such a description is not sufficient for our study, since the preferable independent variable in the inflationary era is the e-foldings number, N. If we are to study the inflationary dynamics of a scalar- Einstein-Gauss-Bonnett model, we need to transform the above relations with respect to the e-foldings numbers, much in the same manner as [305] reconstruct the f (R) inflationary models. At first, we consider a given Hubble expansion rate for the inflationary era, as a function of time, H = H(t); the e-foldings number is defined as

Z t f N = H(t)dt ,(294) ti where ti is the initial and t f the final moments of inflation -usually considered −36 −32 as ti = 10 and t f = 10 [160, 255, 258]. Considering a given initial moment −36 for inflation, ti ∈ [0, 10 ], and an unspecified final moment, t, the e-foldings number is obtained via eq. 294 as a function of time, N = N(t); supposing this function is reversible, time is also given as a function of the e-foldings number, t = t(N). Consequently, the first- and the second-order derivatives with respect to time, as transformed into first- and second-order derivatives with respect to the e-foldings number, as d dN d d = = H(N) and dt dt dN dN (295) d2 dN 2 d2 dN dH d d2 dH d = + = H(N)2 + H(N) . dt2 dt dN2 dt dN dN dN2 dN dN Since the scalar field, χ = χ(t) is a function of time, its potential, V˜ (χ) = V˜ (χ(t)), and the Lagrange multiplier, λ = λ(t), the coupling function, h(χ) = h(χ(t)), as well as the Ricci scalar, the Gauss-Bonnet invariant and the function E(R, χ) = ER(t), χ(t) are also functions of time. As a result, they can all be rewritten with respect to the e-foldings number. Furthermore, the functions Qi(χ) are also transformed, taking the following forms

2 0 2 0 Qa(N) = 8H(N) h (N) , Qb(N) = 16H(N) h (N) ,(296) 2 0 0 Qc(N) = 0 , Qd(N) = 0 , Qe(N) = 32H(N) H (N)h (N) , 2 00 0 0 2 0 Q f (N) = −16 H(N) h (N) + H(N)H (N)h (N) − H(N) h (N) and 2 0 Qt(N) = 1 + 8H(N) h (N) , 110 the ghost-free einstein-gauss-bonnet theory

where tones stand for derivatives with respect to the e-fodlings number -as already stated. In the same manner, we may redeﬁne the wave speed for the gravitational waves,

Q (N) 8H(N)2h00(N) + H(N)H0(N)h0(N) − H(N)2h0(N) 2 = − f = + cT 1 1 2 0 . 2 + Qb(N) 1 + 8H(N) h (N) (297) The next step is to express the slow-roll parameters, ei, with respect to the e-foldings number; they are easily computed to be

H0(N) χ00(N) H0(N) e (N) = , e (N) = + = 0 , 1 H(N) 2 χ0(N) H(N) 1 F0(N) 1 E0(N) e (N) = = 0 , e (N) = 3 2 F(N) 4 2 E(N) (298) Q (N) 4H(N)h0(N) ( ) = a = e5 N 2 0 and H(N) 2 + Qb(N) 1 + 8H(N) h (N) 0 0 00 Q0 (N) H(N) 16H (N)h (N) + 8H(N)h (N) ( ) = t = e6 N 2 0 . Qt(N) 1 + 8H(N) h (N)

Through these, the spectral indices and the tensor-to-scalar ratio are directly calculated with respect to the e-foldings number, using eq. (292) and (293). What remains is to apply a specific coupling function, h(χ), as well as a Hubble expansion rate for the cosmological background of the inflation, so as to calculate the spectral indices and the tensor-to-scalar ratio and compare our results with that of the Planck and BICEP2/Keck-Array observations. We consider two cases for each; as for the coupling function, we stick to the standard exponential and power-law forms, while the expansion of the FRW background is kept simple, at first with a flat DeSitter vacuum, and finally with a flat quasi-DeSitter vacuum.

5.2 the desitter expansion

The DeSitter Hubble rate is constant and positive, signifying an unchanged exponential growth of the scale factor. Due to this, the Hubble rate is not a function of time, and thus not a function of the e-foldings number too

H(t) = H(N) = H0 .(299)

Time and e-foldings number are connected through the simple linear relation

N t = .(300) H0 As a result, the the Ricci scalar and the Gauss-Bonnet invariant are both constant.

2 4 R = 12H0 and G = 24H0 (301) 5.2 the desitter expansion 111

Finally, the scalar ﬁeld, given by eq. (221), takes the following form

µ2 χ(N) = N .(302) H0 Using, eqs. (299), (300) and (302) and a coupling function, we are able to complete the transformation from the time-domain to the e-foldings number.

5.2.1 An exponential coupling function, h(χ) = aebχ

We assume that the coupling function is exponential,

h(χ) = aebχ ,(303) where a and b are real constants, to be used as free parameters later. Using eqs. (302) and (300), we can write the coupling function ﬁrst as function of time,

2 h(t) = aebµ t ,(304) and then as a function of the e-foldings number,

2 bµ t h(N) = ae H0 .(305)

From here, using eq. (225), we may derive the potential as a function of the e- foldings number, 2 bµ N ˜ 2 2 4 H0 V(N) = 8ab H0 µ e ,(306) as well as the Lagrange multiplier,

bµ2 2 2 H N λ(N) = 8ab H0 e 0 .(307)

From the eqs. (296), we derive the Qi functions with respect to the e-foldings number, as

bµ2 bµ2 2 2 H N 2 H N Qa(N) = 8abH0 µ e 0 , Qb(N) = 16abH0µ e 0 ,

Qc(N) = 0 , Qd(N) = 0 , Qe(N) = 0 , (308)

bµ2 bµ2 2 2 H N 2 H N Q f (N) = 16abµ (H0 − bµ )e 0 and Qt(N) = 1 + 8abH0µ e 0 , while the wave-speeds are

bµ2 2 4 H N 1 + 8abH µ e 0 c2 = 1 and c2 = 0 ,(309) A T bµ2 2 H N 1 + 8abH0µ e 0 usign eqs. (288) and (297). 112 the ghost-free einstein-gauss-bonnet theory

The function E(R, χ) is writter with respect to the e-foldings number as

bµ2 N 2 2 2 4 H0 bµ 96a b H0 e 2 N E(N) = − 8abH e H0 .(310) bµ2 0 2 H N 1 + 8abH0µ e 0 Using eqs. (298), (299), (305) and (310), we obtain the slow-roll parameters of the ﬂat DeSitter case with an exponential coupling function, as

e1(N) = 0 , e2(N) = 0 , e3(N) = 0 ,

3bµ2 2bµ2 N N bµ2 3 4 4 4 H0 2 3 3 2 H0 3072a b H0 µ e 384a b H0 µ e 3 2 H N − + − 0  2 bµ2 8ab H0µ e bµ2 N N H 2 H 16abH µ2e 0 +2 16abH0µ e 0 +2 0

e4(N) =  2bµ2  N bµ2 (311) 192a2b2 H4e H0 N 0 − 2 2 H0 2  bµ2 8ab H0 e  N 2 H 16abH0µ e 0 +2

bµ2 bµ2 2 H N 2 4 H N 4abH µ e 0 8ab µ e 0 e (N) = 0 and e (N) = . 5 bµ2 6 bµ2 2 H N 2 H N 1 + 8abH0µ e 0 1 + 8abH0µ e 0 The above allow us to calculate the spectral indices, from eqs. (292),   1 1 1 n = 1 + bµ2 − + − and S  bµ2 bµ2  N N H0 2 2 H0 2 2 H0 8abH0 µ e + H0 H0 − 4aH0 (3H0 − 2bµ ) e ! 2 1 − 2bµ bµ2 1 N 2 H 8abH0µ e 0 +1 nT = H0 (312)

and the tensor-to-scalar ratio, from eq. (293), r bµ2 2 2 H N bµ2 − 0 2 + N abµ bµ H0 8abe H0µ 1 r = 16e H0 .(313) 2 3/2 bµ N 1 + 8ab2e H0 µ4

In order to examine the viability of the model so far, we need to take values for the spectral index nS and the tensor-to-scalar ratio r for the parameters H0, a, b and µ at the end of inﬂation (for N ∈ [50, 60]) and compare these values to the observational results of the Planck collaboration [9, 13] and the BICEP2/Keck- Array [7]. The latest results from Planck, give nS = 0.9649 ± 0.0042 within a 68% signiﬁcance and r < 0.064, while BICEP2/Keck-Array restricts it below 0.07. 5.2 the desitter expansion 113

The reconstructed spectral indices and tensor-to-scalar ratio do not match the observations for a speciﬁc set of values for their parameters. More speciﬁcally, the values of nS and r do not depend on the choice of a, so we set it equal to one for simplicity; they also do not depend on the number of e-foldings, so N = 50 and N = 60 are used in the same manner. They depend on H0, b and µ, 27 12 though; considering that H0 ∼ 10 and setting µ = 10 , we get b = 35.6 so that nS = 0.9644 (Planck’s previous result) and b = 1000 so that r → 0. Figures (10), (11) and especially (12) are decisive on the subject.

1.00 0.10

0.99 0.08

0.98 0.06 s r n

0.97 0.04

0.96 0.02

0.95 0.00 1025 1026 1027 1028 1029 1030 1025 1026 1027 1028 1029 1030

H0 H0

Figure 10: The spectral index nS on the left and the tensor-to-scalar ratio r on the right, 12 with respect to H0, for N = 50 , a = 1 and µ = 10 ; the images are identical for N = 60 and any other values of a. The different colours correspond to different values of b, varying from b = 1 (the blue curve) to b = 1000 (the darker green curve). The horizontal dark red line stands for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, according to Planck 2018 results; the horizontal black line sets the limit r = 0.064 from the same results, while the dashed black an older upper boundary of r = 0.07 from the BICEP2/Keck-Array.

1.00 0.10

0.99 0.08

0.98 0.06 s r n

0.97 0.04

0.96 0.02

0.95 0.00 1010 1011 1012 1013 1014 1010 1011 1012 1013 1014 1015 1016 Μ Μ

Figure 11: The spectral index nS on the left and the tensor-to-scalar ratio r on the right, with respect to µ, for N = 50 and a = 1 ; the images are identical for N = 60 and any other values of a. The different colours correspond to different values 26 29 of H0 and b, varying from H0 = 10 and b = 1 (the blue curve) to H0 = 10 and b = 1000 (the darker green curve). The horizontal dark red line stands for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, according to Planck 2018 results; the horizontal black line sets the limit r = 0.064 from the same results, while the dashed black an older upper boundary of r = 0.07 from the BICEP2/Keck-Array. 114 the ghost-free einstein-gauss-bonnet theory

1.00 0.20

0.99

0.15

0.98 s

0.97 r 0.10 n

0.96

0.05

0.95

0.94 0.00 1 10 100 1000 104 100 104 106 108 1010 1012 b b

Figure 12: The spectral index nS on the left and the tensor-to-scalar ratio r on the right, with respect to b, for N = 50 , a = 1 and µ = 1012 ; the images are identical for N = 60 and any other values of a. The different colours correspond to different 26 29 values of H0, varying from H0 = 10 (the blue curve) to H0 = 10 (the darker green curve). The horizontal dark red line stands for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, according to Planck 2018 results; the horizontal black line sets the limit r = 0.064 from the same results, while the dashed black an older upper boundary of r = 0.07 from the BICEP2/Keck-Array.

5.2.2 A power-law coupling function, h(χ) = aχb

Afterwards, we assume that the coupling function is a simple power law,

h(χ) = aχb ,(314)

where a and b are real constants, to be used as free parameters later. Using eqs. (302) and (300), we can write the coupling function ﬁrst as function of time,

b h(t) = a µ2t ,(315)

and then as a function of the e-foldings number,

b µ2 h(N) = a N .(316) H0

From here, using eq. (225), we may derive the potential as a function of the e- foldings number, 2 b 4 µ 8a(b − 1)bH0 H N V˜ (N) = 0 ,(317) N2 as well as the Lagrange multiplier,

2 b−2 2 µ λ(N) = 8a(b − 1)bH0 N .(318) H0 5.2 the desitter expansion 115

From the eqs. (296), we derive the Qi functions with respect to the e-foldings number, as

2 b 2 b 3 µ 2 µ 8abH0 H N 16abH0 H N Q (N) = 0 , Q (N) = 0 ,(319) a N b N Qc(N) = 0 , Qd(N) = 0 , Qe(N) = 0 ,

2 b 2 b 2 µ 2 µ 16abH0 (N + 1 − b) H N 8abH0 H N Q (N) = 0 and Q (N) = 1 + 0 , f N2 t N while the wave-speeds are

2 b 8a(b − 1)bH2 µ N + N2 2 2 0 H0 cA = 1 and and cT = ,(320) 2 b 8abH2N µ N + N2 0 H0 using eqs. (288) and (297). The function E(R, χ) is writter with respect to the e-foldings number as

b− µ2 2 2 96a2b2 H4 N 2 b−2 0 H0 2 µ E(N) = − 8a(b − 1)bH0 N .(321) 2 b−1 H 1 + 8abH µ2 µ N 0 0 H0

Using eqs. (298), (299), (316) and (321), we obtain the slow-roll parameters of the ﬂat DeSitter case with a power-law coupling function, as e1(N) = 0 , e2(N) = 0 , e3(N) = 0 , (322) b− b− 2 3 4 2 2 3 3 3 4 4 µ 2 2 3 2 µ N 3072a (b−1)b H0 µ N 192a b (2b−2)H0 µ 2 b−3 H0 H0 2 µ − 2 + b−1 − 8a(b − 2)(b − 1)bH0µ N 2 b−1 µ2 N H0 2 µ 2 + 16abH0µ N +2 16abH0µ H 2 H0 0 e (N) = , 4 2 2b−2 4 µ ! 192a2b2 H N 2 b−2 0 H0 − ( − ) 2 µ 2 2 b−1 8a b 1 bH0 H N 2 µ 0 16abH0µ N +2 H0 2 b−1 2 b−2 4abH µ2 µ N 8a(b − 1)bµ4 µ N 0 H0 H0 e5(N) = and e6(N) = . 2 b−1 2 b−1 1 + 8abH µ2 µ N 1 + 8abH µ2 µ N 0 H0 0 H0 116 the ghost-free einstein-gauss-bonnet theory

The above allow us to calculate the spectral indices, from eqs. (292),

b − 1 nS = × 2 b 2 b 2 b N 8abH2 µ N + N N b 12aH2 µ N − 1 + 1 − 8a(b − 1)bH2 µ N 0 H0 0 H0 0 H0 (323) " 2 2b 2 b! 2 2 4 µ 2 2 µ 64a (b − 2)b H0 N + N b − 2 − 24abH0 N − H0 H0

2 b 2 b ! !# 2 µ 2 µ − 16abH0 N N b 6aH0 N − 1 + 2 H0 H0 + 1 and (324)

2 b 16ab(b − 1)H2 µ N 0 H0 nT = (325) 2 b N 8abH2 µ N + N 0 H0

and the tensor-to-scalar ratio, from eq. (293),

2 b 2 µ abH0 (−b + N + 1) H N r = 0 16 3/2 .(326)  2 b  2 µ 2 b 8a(b−1)bH N +N2 2 µ 0 H0 N 8abH N + N 0 H0  2 b  N 8abH2 µ N +N 0 H0

This case of the flat DeSitter vacuum is much more preferable, since there are values of the parameters, for which the values of the spectral index for scalar perturbations and the tensor-to-scalar ratio coincide with their respective observables. Again, we set N = 50 (or N = 60) to indicate the end of the inflationary era, and it is easy to see that the values of H0, a and µ do not matter; as a result, 26 27 we choose a = µ = 1 for simplicity and H0 = 10 (or H0 = 10 ). The tensor-to- scalar ratio is constantly close to zero, while the spectral index coincides with the Planck dat only for µ ∼ 4. Namely, nS = 0.9644 only for b = 3.78 when N = 50, or b = 4.136 for N = 60 - for the same values, r ∈ [10−50, 10−20]. The following Figure (13) proves this. As a result, a power-law coupling function in a flat deSitter background may generate a viable inflationary model, only under the strict assumption of h(χ) ∼ χ4. 5.3 the quasi-desitter expansion 117

1.00

7.´10-26

0.98 6.´10-26

5.´10-26

0.96

4.´10-26 s n r

26 0.94 3.´10-

2.´10-26

0.92

1.´10-26

0 0.90 3.0 3.5 4.0 4.5 5.0 3.0 3.5 4.0 4.5 5.0 Μ Μ

Figure 13: The spectral index nS on the left and the tensor-to-scalar ratio r on the right, with respect to b, for N = 50 , a = 1 and µ = 1012 ; the images are identical for N = 60 and any other values of a. The different colours correspond to different 26 29 values of H0, varying from H0 = 10 (the blue curve) to H0 = 10 (the darker green curve). The horizontal dark red line stands for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, according to Planck 2018 results; the horizontal black line sets the limit r = 0.064 from the same results, while the dashed black an older upper boundary of r = 0.07 from the BICEP2/Keck-Array.

5.3 thequasi-desitter expansion

The quasi-DeSitter Hubble rate is linear function of time, in the form

H(t) = H0 − H1t .(327)

Integrating eq. (327) with respect to time, we obtain H N = H t − 1 t2 , 0 2 and solving with respect to time, we may write the latter with respect to the e-foldings number as q 2 H0 ± H0 − 2H1N t = .(328) H1 As a result, the Hubble rate with respect to the e-foldings number becomes q 2 H(N) = ± H0 − 2H1N ,(329) while the Ricci scalar and the Gauss-Bonnet are written as

2 6H1 2 2 R = 12 H0 − 2H1N ∓ q and G = 24 H0 − 2H1N H0 − H1(2N + 1) 2 H0 − 2H1N (330) Finally, we also express the scalar ﬁeld of eq. (221) with respect to the the e- foldings number, q 2 H0 ± H0 − 2H1N χ(N) = µ2 .(331) H1 118 the ghost-free einstein-gauss-bonnet theory

As in section IV, the eqs. (299), (300) and (302) and a coupling function allow us to transfer the analysis from the time-domain to the appropriate inﬂationary environment. Again, we shall use the same form of the coupling functions

5.3.1 An exponential coupling function, h(χ) = aebχ

At ﬁrst, we shall consider the exponential coupling function eq. (??), that is expressed with respect to the e-foldings number as

√ b H ± H2− H N 0 0 2 1 h(N) = ae H1 .(332)

From here, using eq. (225), we may derive the potential as a function of the e- foldings number,

 √  bµ2 H ± H2− H N √ 0 0 2 1  H   H ± H2−2H N 16ab2µ2 H H2−2H N e 1  2 0 0 1 1( 0 1 ) bµ ± s   H1 √   H2± H N+ H2− H N   0 2 1 0 2 1  ˜ 2 4 2 V(N) = 8ab µ H0 − 2H1N e − q 2 2H1 H0 − 2H1N − , r q 2 2 2 κ H0 − 2H1 3 H0 − 2H1N + N (333)

as well as the Lagrange multiplier,

 √  bµ2 H ± H2− H N 0 0 2 1 r  √ 2 2 H   H ± H2−2H N 16ab H1 (2H1+1)(H −2H1 N)e 1  ± 2 0 0 1 − 0  bµ H 2   1 (2H1+1)µ    2 2 λ(N) = 8ab H0 − 2H1N e ± H1 ± 2 q . 2 4 2 κ µ H0 − 2H1N (334)

The Qi functions with respect to the e-foldings number are derived from the eqs. (296),

Qa(N) = 0 , Qb(N) = 0 , Qc(N) = 0 , Qd(N) = 0 , (335)

Qe(N) = 0 , Q f (N) = 0 , Qt(N) = 1 , (336)

while the wave-speeds are

2 2 cA = 1 and cT = 1 , (337) 5.3 the quasi-desitter expansion 119

usign eqs. (288) and (297). Interestingly, both the Qi functions and the wave- speeds have a trivial form in the case of the quasi-deSitter expansion; this triviality is independent of the coupling function, as we see later, and should be attributed to this speciﬁc FRW background. The function E(R, χ) with respect to the e-foldings number takes the form

H1 E(N) = ± 2 q − 2 4 2 κ µ H0 − 2H1N  √  bµ2 H ± H2− H N 0 0 2 1 √ r  2 2 2 H   H0± H −2H1 N 16ab H1 (2H1+1)(H −2H1 N)e 1   2 0 ± 0  bµ H 2   1 (2H1+1)µ    2 2 − 8ab H0 − 2H1N e . (338)

Using eqs. (298), (327), (332) and (338), we obtain the slow-roll parameters of the ﬂat quasi-DeSitter case with an exponential coupling function. Interestingly, the ﬁve of them take the following trivial form, that seems independent of the coupling function, while the fourth has a long and complex form depending on the coupling function, H ( ) = − 1 ( ) = ( ) = e1 N 2 , e2 N 0 , e3 N 0 , (339) H0 − 2H1N e4(N) = εexp(N, H0, H1, a, b, µ) , e5(N) = 0 and e6(N) = 0 .

Similarly, the spectral indices and the tensor-to-scalar ratio are also long and complex functions of the e-foldings number, the mass µ and the model parameters, H0 and H1 due to the expansion rate and a and b due to the coupling function; thus we do not present them in close form. What is interesting to note is that the spectral indices and the tensor-to-scalar ratio yield the same values independently of which case of eq. (328) we will use. Again, we perform comparisons using the observable values for nS and r obtained by the Planck with their latest data [13], along with [9] and [7]. As we stated before, the spectral index of the scalar modes must within the interval [0.9607, 0.9691] with a 68% significance and mean nS = 0.9649; the tensor-to-scalar mode, on the other, is restricted below 0.1 by [7] and below 0.07 by [9], while [13] restricts further as r < 0.064. In our case, the parameters a and b, as well as the mass µ of the scalar field seem to play no part in the value of the spectral indices or the tensor-to-scalar ratio; as a result, we consider them equal to unity (a = b = µ = 1), so that the analysis is simplified and focused on the rest of the parameters. The e-foldings number is chosen N = 50 and N = 60, so as to indicate the end of inflation, but 14 this also does not alter the results. As for the expansion rate, given that H0 ≥ 10 26 14 27 for H1 ≈ 10 (or that H0 ≥ 510 for H1 ≈ 10 ), the spectral index approaches 12 15 unity, restricting our choices. We consider H0 to be in the interval [10 , 10 ] and 120 the ghost-free einstein-gauss-bonnet theory

26 29 H1 in the respective interval [10 , 10 ], where we can equalise the spectral index of our model to the observable value, as we can see in Figure (14).

1.00 1.00

0.99 0.99

0.98 0.98 s s n n

0.97 0.97

0.96 0.96

0.95 0.95 1013 1014 1015 1016 1017 1013 1014 1015 1016 1017

H0 H0

Figure 14: The spectral index nS with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1. The image is identical for N = 60 and any other values of a, b and µ. The blue, cyan, green and darker green curves correspond to different values of H1 and H0, respectively. The horizontal dark red line stands for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, according to Planck 2018 results .

For the majority of these cases, the tensor-to-scalar ratio is close to zero, as we can see in Figure (15).

0.10 0.10

0.08 0.08

0.06 0.06 r r

0.04 0.04

0.02 0.02

0.00 0.00 1013 1014 1015 1016 1017 1013 1014 1015 1016 1017

H0 H0

Figure 15: The tensor-to-scalar ratio with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1. The image is identical for N = 60 and any other values of a, b and µ. The blue, cyan, green and darker green curves correspond to different values of H1 and H0, respectively. The horizontal dashed black line sets the limit r < 0.07, while the horizontal black line the limits r < 0.064, according to Planck 2015 and Planck 2018 results, respectively.

27 As an example, choosing N = 50 (or N = 60) and H1 = 10 , then for H0 = 14 4.91375 10 , we have nS = 0.9644 and r = 0.0282787, which satisfy even the latest data of the Planck collaboration. What we need to notice is that these two parameters (H0 and H1) need careful ﬁne-tuning and cannot diverge a lot from this set of values, otherwise the model collapses before the data. 5.3 the quasi-desitter expansion 121

5.3.2 A power-law coupling function, h(χ) = aχb

At ﬁrst, we shall consider the power-law coupling function eq. (??), that is expressed with respect to the e-foldings number as

√ b H ± H2− H N 0 0 2 1 h(N) = ae H1 .(340)

From here, using eq. (225), we may derive the potential as a function of the e- foldings number,

˜ 4 2 V(N) =8a(b − 1)bµ H0 − 2H1N ×  √ b−1 b−2 2 2 ! µ H0± H ±2H1 N q 16abH µ2 H2 − 2H N 0  2 2 1 0 1 H1   µ H0 ± H0 − 2H1N   ±  −  H r q   1 2 2   H0 ± 2H1 H0 − 2H1N + N 

q 2 2H1 H0 − 2H1N − , r q 2 2 2 κ H0 − 2H1 3 H0 − 2H1N + N (341) as well as the Lagrange multiplier,

2 λ(N) = ± 8a(b − 1)b H0 − 2H1N ×  √ b−1 b−2 2 2 ! q µ H0± H −2H1 N q 16abH (2H + 1) H2 − 2H N 0  2 2 1 1 0 1 H1   µ H0 ± H0 − 2H1N   ±  −  2   H1 (2H1 + 1)µ   

2H1 − q . 2 4 2 κ µ H0 − 2H1N (342)

The Qi functions with respect to the e-foldings number are derived rom the eqs. (296), while the wave-speeds are given from eqs. (288) and (297). As we noted before, these have the same trivial form as in eqs. (335) and (337). 122 the ghost-free einstein-gauss-bonnet theory

The function E(R, χ) with respect to the e-foldings number takes the form

2H1 2 E(N) = ∓ q − 8a(b − 1)b H0 − 2H1N × 2 4 2 κ µ H0 − 2H1N b−  √ !b−1  2 µ2 H ± H2−2H N q 2 0 0 1  2 2 16abH1 H0 − 2H1N H   µ H0 ± H − 2H1N 1   0   ± q  .  H1 µ2 H2 − H −H2 + H N + N   0 2 1 0 2 1 

(343)

Using eqs. (298), (299), (340) and (343), we obtain the slow-roll parameters of the ﬂat quasi-DeSitter case with an exponential coupling function. Except for the fourth one, which is a long and complex expression,

e4(N) = ε pow(N, H0, H1, a, b, µ) ,(344)

the rest are given by their trivial expressions of eqs. (339). The spectral indices and the tensor-to-scalar ratio have the same form as in the case of the exponential coupling function, presented above. Again, we perform comparisons using the observable values for nS and r obtained by the Planck with their latest data [13], along with [9] and [7]. As we stated before, the spectral index of the scalar modes must within the interval [0.9607, 0.9691] with a 68% significance and mean nS = 0.9649; the tensor-to-scalar mode, on the other, is restricted below 0.1 by [7] and below 0.07 by [9], while [10, 13] restricts further as r < 0.064. In our case, the parameters a and b, as well as the mass µ of the scalar field seem to play no part in the value of the spectral indices or the tensor-to-scalar ratio; as a result, we consider them equal to unity (a = b = µ = 1), so that the analysis is simplified and focused on the rest of the parameters. The e-foldings number is chosen N = 50 and N = 60, so as to indicate the end of inflation, but 14 this also does not alter the results. As for the expansion rate, given that H0 ≥ 10 26 14 27 for H1 ≈ 10 (or that H0 ≥ 510 for H1 ≈ 10 ), the spectral index approaches 12 15 unity, restricting our choices. We consider H0 to be in the interval [10 , 10 ] and 26 29 H1 in the respective interval [10 , 10 ], where we can equalise the spectral index of our model to the observable value, as we can see in Figure (16). For the majority of these cases, the tensor-to-scalar ratio is close to zero, as we can see in Figure (17). 27 11 Setting N = 50 (or N = 60) and H1 = 10 , then for H0 = 8.43822 10 (or 12 H0 = 10 ), we have nS = 0.9644 and r = 0.0400002 (or r = 0.0333335), that match the latest data of the Planck collaboration. Again, these two parameters (H0 and H1) need careful fine-tuning and cannot diverge a from these values, or the model collapses before the data. 5.3 the quasi-desitter expansion 123

1.00 1.00

0.99 0.99

0.98 0.98 s s n n

0.97 0.97

0.96 0.96

0.95 0.95 1013 1014 1015 1016 1017 1013 1014 1015 1016 1017

H0 H0

Figure 16: The spectral index nS with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1. The image is identical for N = 60 and any other values of a, b and µ. The blue, cyan, green and darker green curves correspond to different values of H1 and H0, respectively. The horizontal dark red line stands for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, according to Planck 2018 results [10, 13].

0.10 0.10

0.08 0.08

0.06 0.06 r r

0.04 0.04

0.02 0.02

0.00 0.00 1013 1014 1015 1016 1017 1013 1014 1015 1016 1017

H0 H0

Figure 17: The tensor-to-scalar ratio with respect to H0 the left, and to H1 in the right, for N = 50 and a = b = µ = 1. The image is identical for N = 60 and any other values of a, b and µ. The blue, cyan, green and darker green curves correspond to different values of H1 and H0, respectively. The horizontal dashed black line sets the limit r < 0.07, while the horizontal black line the limits r < 0.064, according to Planck 2015 and Planck 2018 results, respectively.

5.3.3 The coupling-free inﬂation of a quasi-DeSitter background

Our study for a linear expansion rate, in the form of eq. (327), yield the same results for the spectral indices and the tensor-to-scalar ratio, independently of the coupling function used. In greater depth, the results of both cases studied were similar, while the Qi functions, the wave speeds and ﬁve out of six of the slow-roll parameters proved to be the same; in these we should also include the tensor-to-scalar ratio that takes the same form in both cases, H = 1 r 4 2 .(345) H0 − 2H1N On the other hand, the sixth slow-roll parameter and the spectral indices -though being too complex to examine thoroughly-present great similarity. 124 the ghost-free einstein-gauss-bonnet theory

All these evidence allow us to believe this “coupling-free” state to be a feature of the Hubble rate in the background, rather that of the specific Einstein- Gauss-Bonnet theory used - e.g. of the coupling function chosen. In fact, the only parameters in need of fine-tuning for the model to be compatible to the observations, were the H0 and H1 constants of the Hubble rate. The figures ?? and ?? present the chance of coincidence for the nS and r according to the quasi-deSitter reconstruction, for any of the coupling functions and for varying of H0 and H1, respectively. The values of the two parameters are proved to be connected by a specific relation, log(H1) − log(H0) = l − ci ,(346)

where l the difference of the H0 and H1 orders and codd ' log(2.5) = 0.39739 when l is an odd number, and ceven ' log 8.5 = 0.92942 when it is an even number, so that the values of nS and r are both compatible to the observables.

0.12

0.10

0.08

26 N=50 , H1=10 , a=b=Μ=1

Tensor Ratio 0.06 - 27 to N=50 , H1=10 , a=b=Μ=1 -

28 Scalar N=50 , H1=10 , a=b=Μ=1

0.04

0.02

0.00 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Spectral Index

Figure 18: A parametric plot of the tensor-to-scalar ratio (vertical axis) over the spectral index (horizontal line) for N = 50, a = b = µ = 1 and varying H0 for speci- ﬁed H1. The vertical red line stands for for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, while the horizontal dotted black line sets the limit r < 0.1, the horizontal dashed black line sets the limit r < 0.07 and the horizontal black line the limits r < 0.064, according to Planck 2015 and Planck 2018 results, respectively.

5.4 conclusions

The realisation of the inflationary era in the context of the ghost-free Gauss- Bonnet gravity is proved viable for a universe expanding with a linear Hubble rate, regardless of the coupling of the scalar field to the Gauss-Bonnet invariant. However, a careful fine-tuning is needed for the constants of the Hubble rate; if they happen to fulfill the eq. (346), then the realised model is viable and compatible to the observations. On the contrary, a typical deSitter evolution is depending 5.4 conclusions 125

0.12

0.10

0.08

14 N=50 , H0=2 10 , a=b=Μ=1

Tensor Ratio 0.06 - 14 to N=50 , H0=5 10 , a=b=Μ=1 -

15 Scalar N=50 , H0=2 10 , a=b=Μ=1

0.04

0.02

0.00 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 Spectral Index

Figure 19: A parametric plot of the tensor-to-scalar ratio (vertical axis) over the spectral index (horizontal line) for N = 50, a = b = µ = 1 and varying H1 for speci- ﬁed H0. The vertical red line stands for for nS = 0.9649, while the horizontal dashed red lines for the limits of its conﬁdence interval, while the horizontal dotted black line sets the limit r < 0.1, the horizontal dashed black line sets the limit r < 0.07 and the horizontal black line the limits r < 0.064, according to Planck 2015 and Planck 2018 results, respectively

highly on the nature of the coupling, with the exponential coupling function to fail completely and the power-law coupling function to generate a viable model, compatible with the observations only for a small interval of values for the mass µ -approxiamtely between 4 and 4.2. This illustration comes after a series of similar cases (see refs. [53, 54, 61] for example) that demonstrate the capability of the Gauss-Bonnet gravity, coupled with a scalar ﬁeld, to generate a realistic inﬂationary model.

Part IV

PHASESPACEANALYSISOFALTERNATIVE COSMOLOGIES

Given a theory of gravity and speciﬁc conditions, a dynamical model can be constructed to account for its viability. 1. Initially, the intracting two-ﬂuids description of dark matter and dark energy is studied, for both classical and Loop Quantum Cos- mology cases. 2. Finally, a k-essence f (R) theory is presented and parametrised as an autonomous dynamical system, whose phase space is examined.

THEINTERACTINGFLUIDSDESCRIPTIONOFDARK 6 MATTERANDDARKENERGY

As stated many times so far, the current perception of the Universe is that of a flat space filled with three components, interpreted as fluids. The two of them, namely luminous matter -either baryonic or relativistic- and dark matter, extend to approximately 32 % of the Universe and share features of typical matter fields P and are usually interpreted as ideal fluids, with barotropic index w = ≥ 0. ρ The third, usually referred to as dark energy, dominates the rest ∼ 68 % of the Universe and is usually formulated as an ideal fluid as well with barotropic index w ' −1 [52, 130, 243, 395]. Yet, our understanding of the nature of dark matter and dark energy is so few that attributing them features of the simplest ideal gases is hardly but a good approximation. Several ideas were proposed that, despite respecting their status as ideal fluids, so the the flat FLRW background shall be preserved, referred to them as exotic fluids, presenting completely different equations of state. For example, refs. [192, 340, 370] and more recently [55–57, 104, 188, 214, 215, 334] propose the idea of superfluid dark matter, such as Bose-Einstein condensates, while ref. [75] stretches that gravitational waves probe for the superfluid nature of dark matter and refs. [4, 189] points to superdense He-3 and He-4 Bose-Einstein condensates as dark matter candidates. In a similar manner, the dark energy can also be interpreted as an exotic type of fluid, with highly nonlinear equation of state. As early as 2001 and refs. [62, 63, 93, 129, 140, 159, 162], the polytropic and the Chaplygic gases have been proposed as plausible sources for the late-time accelerating expansion, even for a unified theory of dark matter and dark energy. Some of these claimed that phantom dark energy is produced by this approximation, since the effective barotropic index was sufficiently lower than −1 [5, 109, 124, 125, 128, 152, 161, 170, 182, 200, 240, 262, 275, 354, 400–402, 413]. Models where the Chaplygin gas was coupled to modified gravity terms have been considered, including the f (R), f (G) and f (T ) theories (see refs. [60, 102, 138, 139, 209–211, 272, 273, 345, 350]), the Loop Quantum Cosmology (refs. [201, 414] inter alia), or even the holographic Universe (see [364, 372, 391]). The unification with dark matter via this formulations has been proved viable [64, 192, 240, 325, 329, 333, 340, 402, 403, 406, 413], and even the unification with early-time accelerating expansion [327, 414]. Furthemore, a wide variety of observational results support both the case for a Chaplygic gas, as well as the interaction between dark matter and dark energy under this formulation [92, 97, 117, 237, 238, 241, 343, 344, 356, 367, 394, 396, 404, 405].

129 130 the interacting fluids description of dark matter and dark energy

In this chapter, we conduct the analysis of two dynamical systems founded on the assumption of superfluid dark matter interacting with a generalised Chap- lygin gas dark energy. The first of the two models treats the classical case of a flat FLRW Universe, while the latter assumes corrections in the sense of the Loop Quantum Cosmology (LQC). Both cases reveal an interesting phase space, but not equally interesting results. In the classical case, a stable equilibrium point is reached under circumstances, that behaves very similarly to the observed distri- bution between dark matter and dark energy; on the contrary, in the LQC case, no attractor exists in the phase space, deeming plausible the chance for a finite-time future singularity. Similar studies concerning the approach of dynamical systems in the cosmology can be traced in refs. [217, 218, 325] inter alia, with interacting dark matter and dark energy cosmologies playing a vast role [163, 318, 325, 333, 407]; studies including LQC have also been conducted [201, 329, 414]. The case of future singularities has often discussed, eg. by refs. [95, 134, 219, 319, 329]. The following may be viewed as a continuation of the work in refs. [325, 329].

6.1 thetwo-fluids description

First, we will assume a ﬂat Friedmann-Lemaître-Robertson-Walker space-time, with metric deﬁned in eq. (105) as

3 2 ds2 = −dt2 + a(t)2 ∑ dxi , i=1 a˙ where a(t) is the scale factor and H = is the Hubble rate, both dealing with the a spatial expansion. We further consider that the Universe is ﬁlled with two ﬂuids; the evolution of each is described by a continuity equation in the form

ρ˙i + 3H (ρi + Pi) = Qi ,(347)

where i = dm will denote dark matter and i = de will denote dark energy; furthermore, Qi denotes their interaction. The two continuity equations are accompanied with two equations of state, each one encapsulating the specific features of the fluid. Beginning with dark matter, we investigate the case of superfluidity, hence its euation of state will have the form

4 3 Pdm = κ Γρdm ,(348)

where Γ a parameter specifying the level of superﬂuidity. For the dark energy, next, we will consider a generalised Chaplygin gas that contains a linear and a quadratic term, an inverse term and a logarithmic term as follows,

κρ Λ = −( + ) + 2 de − 2 2 − Pde 1 wd ρde κ Aρde ln 2 κ Bρde 4 ,(349) 3H κ ρde 6.2 the classical case (1): setting up the model 131

where again A, B, Λ and wd are parameters indicating the amplitude of each term. Clearly, setting all of them equal to zero, means that dark energy attains its familiar form of a negative-pressure ideal gas. As for the interaction between the two fluids, we follow a very simple idea. We suppose that the interacting terms, Qi, are (1) of equal magnitude and different sign, and (2) proportional to the Hubble rate and the mass-energy densities of the tow fluids. The first hypothesis is only logical, since interaction fro the two fluids means the transfer of energy from the one to the other. As for the second hypothesis, it is frequently used in the relevant literature, because of its simplicity but also of its reasoning that energy is transferred from one fluid to the other only if it is already “stored” in the first. As a result, the interacting terms have the following structure

Qdm = 3H (c1ρdm + c2ρde) and Qde = −3H (c1ρdm + c2ρde) ,(350) where c1 and c2 are coupling constants. Using eqs. (348) and (350), the continuity equation for dark matter is written as

4 3 ρ˙dm + 3H (1 − c1)ρdm + κ Γρdm − c2ρde = 0 . (351) In the same manner, using eqs. (349) and (350), the continuity equation for dark energy becomes κρ Λ + ( − ) + de − 4 2 − + = ρ˙de 3H c1 wd ρde κAρde ln 2 κ Bρde 4 c2ρdm 0 . (352) 3H κ ρde Finally, using eq. (350), the effective equation of state for the total fluid can be defined as Pdm + Pde we f f = , ρdm + ρde as one could easily imagine, and eventually take the form ρ ρ κρ ρ2 ρ2 = −( + ) de + de de − 2 de + 4 dm we f f 1 wd κA ln 2 κ B κ Γ . ρdm + ρde ρdm + ρde 3H ρdm + ρde ρdm + ρde (353) Eqs. (351) and (352) offer the first two differential equations of our dynamical system, while eq. (353) offers the effective barotropic index, that allows us to compare the results of the phase space analysis with observational data. In order for the system to be complete, we need another differential equation, one governing the evolution of the Hubble rate. This equation is merely the Raychaudhuri equation. Since the latter changes significantly in the classical and the LQC case, we shall introduce each one seperately.

6.2 the classical case (1): setting up the model

In the classical case, the Friedmann equation is taken from eq. (20) and written as κ2 H˙ = − (ρ + ρ + P + P ) ,(354) 3 dm de dm de 132 the interacting fluids description of dark matter and dark energy

and using eqs. (348) and (349), we can rewrite it as

κ κρ Λ ˙ = − + 4 3 − + de − 4 2 − H ρdm κ Γρdm wdρde κAρde ln 2 κ Bρde 4 .(355) 3 3H κ ρde This is the third equation to complete the system. Of course, in order for the system to be simple and autonomous, it is preferable to express it in terms of dimensionless phase space variables, such as the density parameters. The deﬁnition of the phase space variables follows as κρ κρ x = de , x = dm and z = κH2 .(356) 1 3H2 2 3H2 In order to take account for rapid eras (eg. late-time acceleration), we shall not use the cosmic time, but the e-foldings number as the independent variable of the system. The transformation of derivatives from the cosmic time to the e-foldings number is given in eq. (245) in chapter 4. The evolution of the phase space variables with respect to the e-foldings number, N, is given as

dx κ κ 1 = ρ˙ − H˙ , dN 3H de 3H2 dx κ κ 2 = ρ ˙ − H˙ and dN 3H dm 3H2 dz = 2κH˙ . dN Substituting from eqs. (351), (352) and (355) and after simple mathematical manipulations, we obtain the following system

dx Λ Λ 1 = ( − 2) + + 2 − 3 + + 3 2 − + − 3wd x1 x1 3x1x2 9Bx1z 9Bx1z 27Γx1x 2 z 2 2 dN 3z 3x1z 2 − 3A(x1 + x1) ln x1 − (c2x1 + c1x2) , dx Λx 2 = − 2 − 3 2 + ( + ) − − 2 + 3Ax1x2 ln x1 9Bx1x2z 27Bx2z c1x2 c2x1 3wdx1x2 2 dN 3x1z 4 2 2 + 27Γx2z + 3x2 − 3x2 and

dz 2 2 Λ 3 3 = − 3Ax1z ln x1 + 9Bx1z + 3wdx1z + − 27Γx2z − 3x2z . dN 3x1z (357)

In the same manner, the effective equation of state from eq. (353) becomes

9(1 + wd)x2z2 + 27Bx3z3 − 81Γx x3z4 + Λ − 9Ax2z2 ln x = − 1 1 1 2 1 1 we f f 2 .(358) 9x1(x1 + x2)z2 Finally, we know that the system of eqs. (357) is subject to a constraint, arising from the fact that the total matter and energy of the Universe is included. Using 6.3 the classical case (2): working out the model 133 the Friedmann equation from eq. (19) of chapter 1 to express this totality, we may reach the constraint. Expressing equation κ H2 = (ρ + ρ ) 3 dm de with respect to the phase space variables, we may write it as

x1 + x2 = 1 . (359)

That is the Friedmann constraint, proposing that all viable solutions of system (357) must fulfill this if the Universe is to be flat and not containing any other matter field; if x1 + x2 > 1, then the Universe must be closed (with negative spatial curvature), while if x1 + x2 < 1 it must be open (with positive spatial curvature).

6.3 the classical case (2): working out the model

The system (357) is highly non-linear and presents a pole in x1 = z = 0; as a result, solutions cannot contain this case, except for when Λ = 0. Furthermore, the nonlinearity of the system does not allow us to locate analytically any equilibrium point in the general case. The only hint we have towards the existence of an equilibrium point, is that

1 = − × A ∗ 2 ∗ 3 ∗ 2 ∗ 9(x1 ) (x2 ) (z ) ln x1 ∗ 3 ∗ ∗ ∗ ∗ 4 ∗ 2 ∗ 4 ∗ 2 ∗ 3 ∗ 2 c1(x1 ) x2 z − 3c1x1 (x2 ) (z ) + c2(x1 ) z − 3c2(x1 ) (x2 ) (z ) +

∗ 2 ∗ 3 ∗ 2 ∗ 3 ∗ ∗ ∗ 3 + 9wd(x1 ) (x2 ) (z ) − 3(x1 ) x2 z + Λ(x2 ) and c x∗ + c x∗ − 3x∗ = = 1 2 2 1 2 B Γ ∗ 3 ∗ 2 , 27(x2 ) (z ) ∗ ∗ ∗ where x1, x2 and z stand for the respective values in an equilibrium point. This means that parameters B and Γ must be equal, if an equilibrium point is to exist. It is also interesting that the existence of -at least- one equilibrium point does not demand that the parameters take values from specific intervals, though the same may be thought for stability; as long as eqs. (??) hold, so that real -and realistic- for parameters c1, c2, wd and Λ generate real -and realistic- values for a, B and Γ, ∗ ∗ ∗ as well as for x1, x2 and z , no need for fine-tuning exists. To prove so, henceforth we shall assume c1, c2, wd and Λ as exogenously given constants, with values c1 = c2 = 1 and wd = 1/3, without any loss of generality. It is shown that the system (357) contains a stable equilibrium point for a wide variety of A, B and Γ values. In the following we shall investigate the phase space of the system in three specific but quite broad cases. 134 the interacting fluids description of dark matter and dark energy

6.3.1 The generalised dark energy

Assuming that B = Γ = Λ0, the equations take the form

dx 1 =3w (x − x2) + 3x x + 3A(x + x2) ln x − (c x + c x ) , dN d 1 1 1 2 1 1 1 2 1 1 2 dx 2 =3Ax x ln x + (c x + c x ) − 3w x x + 3x2 − 3x and dN 1 2 1 1 2 2 1 d 1 2 2 2 dz = − 3Ax z ln x + 3w x z − 3x z . dN 1 1 d 1 2 In this case, we refer to an ideal fluid dark matter and a generalised fluid dark energy. The analytical tracing of a fixed point is again proved fruitless, due to the persistence of the logarithmic terms. However, we may trace numerically the following two equilibrium points

P1 (0.639858, 0.360142, 0) and P2 (0.122596, 0.877404, 0) .(360)

The first indicates ∼ 64 % of dark energy and ∼ 36 % of dark matter, along with zero expansion of the 3-d space, since H = 0; this is an Einstein static Universe, whose matter fields content assimilates the matter content of ours. The second is again an Einstein static universe, but its matter content appears reversed to that of ours, with ∼ 12 % of dark energy and ∼ 88 % of dark matter. Thus, P2 can clearly be attributed to an earlier phase of the Universe, where matter is still dominant and dark energy is not yet triggered. P1, on the other hand, could quite well describe the late-time evolution of the Universe, if not for the zero expansion.1 To account for the stability of the equilibrium points, we attempt to utilize the Hartman-Grobman theorem and linearise the system (357) around each of the equilibria. In doing so for P1, we attain the linearised system   dξ1      dN  −1.37469 0.919573 0 ξ1  dξ2        =  1.59801 −0.696254 0  ξ  ,(361)  dN     2   dξ3 0 0 −0.223319 ξ3 dN ∗ where ξi are small linear perturbations around the equilibrium values, xi . The eigenvalues of the system are λ1 = −2.29426, λ2 = −0.223319 and λ3 = 0.223319; two of them are real and negative, indicating attraction, while the third is real and positive indicating repellence. As it is obvious from Fig. 20, this should mean that a saddle behaviour appears around the equilibrium, with one or more of the variables repelled away from it; more specifically, since direction v3 = (0, 0, 1) is the eigenvector of a negative eigenvalue, the Hubble rate is expected to approach equilibrium with the the mass-energy densities being repelled from it. However,

∗ ∗ 1 The values of x1 and x2 can be further ﬁxed to approach the observational values, with a more careful choice of c1, c2 and wd. 6.3 the classical case (2): working out the model 135

this is not the case if the equilibrium point P1 is approached from a specific direction, that of the constraint. As can be seen in the first subplot of Fig. 20, all initial conditions for which the constraint is fulfilled -as well as those below the ∗ constraint- lead to the equilibrium. We should also notice that the values of x1 ∗ 2 and x2 yield we f f = −1, due to the choice of wd = 0. For the stability of P2, we have the similar linearised system   dξ1      dN  3.7527 −0.632211 0 ξ1  dξ2        = −1.89242 2.49248 0  ξ  ,(362)  dN     2   dξ3 0 0 −1.86027 ξ3 dN

with eigenvalues λ1 = 4.38491, λ2 = 1.86027 and λ3 = −1.86027. Here, direction v3 = (0, 0, 1) is the only one accociated with a positive eigenvalue, thus the only one indicating an attracting direction, which corresponds to the attraction of the Hubble rate to zero. As for the density parameters x1 and x2 they are both repelled to equilibrium point P1. This case contains an Einstein static Universe attractor, that cannot be considered a viable option, since all observations conclude to an expanding Universe. As a result, other cases must be considered, with the superﬂuidity of the dark matter taken into consideration, so that deSitter or quasi-deSitter attractors may appear.

6.3.2 The superﬂuid dark matter and generalised dark energy

Taking into account the analytical result that B = Γ, so that an equilibrium point is generated, we may consider ﬁrst the case for Λ = 0. Here, the system contains no logarithmic or inverse terms, hence its non-linearity is restricted. However, an analytical quest for equilibrium points is yet fruitless. Numerical estimates claim that -at least- two equilibrium points are located

P1 (0.736157, 0.263844, 2.0503) and P2 (0, 0, −1.57575) .(363)

The first of them corresponds to ∼ 73 % of dark energy, ∼ 26 % of dark matter and a constant but very low Hubble rate, hence it assimilates quite well the current deSitter phase of the Universe. The second of them corresponds to zero matter fields contained in the Universe and a very low but negative Hubble rate, similar to an anti-deSitter shrink of the Universe. The latter does not fulfill the

2 Parameter wd is capable of changing both the the value of we f f in this case, as well as the stability 1 of point P1. The choice wd = , for example, yields the same realistic equilibrium point, with ∗ ∗ 3 x1 = 0.72 and x2 = 0.28, but with we f f = −1.33; furthermore, while variables x1 and x2 are naturally lead to equilibrium, direction v3 = (0, 0, 1) is now unstable and hence the Hubble rate explodes. 136 the interacting fluids description of dark matter and dark energy

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0.8

1.5

0.6 2

z 1.0 x

0.4

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

x1 x1

2.0

1.5

1.0

0.5

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0.0 0.2 0.4 0.6 0.8 1.0

Figure 20: Intersections of the phase space, for c1 = c2 = 1, wd = 0, A = 1, B = Γ = 0 and Λ = 0. The ﬁrst plot corresponds to z = 0 (equilibria P1 and P2), while the second and the third correspond to x2 = 0.360142 and x1 = 0.639858 respectively (equilibrium P1). Blue streamlines stand for trajectories in the phase space slices, red spots mark the equilibrium points and the dashed black line denotes the constrain.

Friedmann constraint and cannot be considered as a viable condition for the Uni- verse. We should note that the effective barotropic index for the P1 is strictly we f f = −1, corresponding to “dark energy”, while for the P2 tends to inﬁnity, another hint for the latter’s unviability. Concerning their stability, we shall again consider small linear perturbations around the equilibria. Given the area of P1, the linearised system is   dξ1      dN  −0.283187 2.95343 0.278386 ξ1  dξ2        =  0.283184 −2.95342 −0.278386 ξ  ,(364)  dN     2   dξ3 5.57028 −11.0108 0.583064 ξ3 dN 6.3 the classical case (2): working out the model 137

∗ where ξi are small linear perturbations around the equilibrium values, xi . The eigenvalues of the system are λ1 = −4.20139, λ2 = 1.54784 and λ3 = 0; the results probe to the existence of a stable, an unstable and a neutral manifold, hence P1 is non-hyperbolic point and the Hartman-Grobman theorem cannot be utilised as was in the previous case. However, the existence -and the fulfillment- of the Friedmann constraint restricts solutions on the surface x1 + x2 − 1 = 0, hence the neutral manifold may not interfere with them. The result, as shown from numerical estimates in Fig. 21, is that initial values for density parameters that fulfill the constraint -or even x1 + x2 > 1- end up normally to the equilibrium point, though the saddle behaviour reappears when the Hubble rate is considered; in the end, it is very hard to consider this viable equilibrium point as stable, and the possible evolution of the system is an explosive behaviour. As for the unviable equilibrium point P2, the linearised system is   dξ1      dN  −0.283187 2.95343 0.278386 ξ1  dξ2        =  0.283184 −2.95342 −0.278386 ξ  ,(365)  dN     2   dξ3 5.57028 −11.0108 0.583064 ξ3 dN with eigenvalues λ1 = −1.5 + 0.866025i, λ2 = −1.5 − 0.866025I and λ3 = 0. The presence of two complex eigenvalues with negative real part should indicate attracting oscillations towards this equilibrium, however the presence of a neutral manifold deems the point a non-hyperbolic one and the Hartman-Grobman theorem again cannot be applied. The non-fulfillment of the Friedmann constraint does not contain the solutions on some surface of the phase space, hence the stability of this equilibrium point cannot be addressed in he typical manner. Furthermore, we can allow A to vary beyond zero, so that the logarithmic terms are also present in the equation of state of the dark energy. Thus, for eg. A = 1, B = Γ = 0.01 and Λ = 0, the system (357) may take the form

dx 1 1 1 = − 3 + 2 + 3 2 + + − − − 0.9x1z 0.9x1z 2.7x1x2z 3x1x2 2 x1 x2 2 , dN 3x1z 3z dx x 2 = − + 2 − 2 − 2 − 3 2 + 4 2 x1 2x2 3x2 2 0.9x1x2z 2.7x2z 2.7x2z and dN 3x1z dz 2 2 1 3 3 =0.9x1z + − 2.7x2z − 3x2z . dN 3x1z This system of differential equations contains the following equilibrium point

P1 (0.687883, 0.312117, 0.88051) ,(366) that is physically meaningful, respects the Friedmann constraint and yields we f f = −1, as expected. Furthermore, it is aggreable with observational results for the current state of the Universe, since it ends up with ∼ 69 % of dark energy, ∼ 31 % 138 the interacting fluids description of dark matter and dark energy

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0.8 2.0

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0.4 1.0

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0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

x1 x1

2.5

2.0

1.5

1.0

0.5

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 21: Intersections of the phase space, for c1 = c2 = 1, wd = 0, A = 0, B = Γ = 0.01 and Λ = 0. The ﬁrst plot corresponds to z = 2.0503 (equilibrium P1), while the second and the third correspond to x2 = 0.263844 and x1 = 0.736157 respectively (equilibrium P1). Blue streamlines stand for trajectories in the phase space slices, red spots mark the equilibrium points and the dashed black line denotes the constrain.

of dark matter and a very low positive Hubble rate, indicating deSitter expansion. Linearising the system in the area of poit P1, we obtain the following   dξ1      dN  −0.943313 1.48448 −0.210739 ξ1  dξ2        =  0.943313 −1.48448 0.210739  ξ  ,(367)  dN     2   dξ3 0.15992 −3.1802 −1.00236 ξ3 dN

with eigenvalues λ1 = −1.71508 + 0.442645i, λ2 = −1.71508 − 0.442645i and λ3 = 0. Again, this equilibrium point is proved a non-hyperbolic, due to the existence of a neutral manifold, hence the Hartman-Grobman theorem should not be applied. However, as we noted before, the Friedmann constraint restricts solutions on the surface x1 + x2 − 1 = 0, where the stability of the equilibrium 6.3 the classical case (2): working out the model 139 point may indeed depend on the other two eigenvalues; these indicate decreasing oscillations, since they are complex with negative real part.

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Figure 22: Intersections of the phase space, for c1 = c2 = 1, wd = 0, A = 0, B = Γ = 0.01 and Λ = 1. The ﬁrst plot corresponds to z = 2.0503 (equilibrium P1), while the second and the third correspond to x2 = 0.263844 and x1 = 0.736157 respectively (equilibrium P1). Blue streamlines stand for trajectories in the phase space slices, the red spot marks the equilibrium points and the dashed black line denotes the constrain.

We observe from Fig. 22 that trajectories fulﬁlling the Friedmann constraint converge to the equilibrium point in the x1 − x2 plane, while the Hubble rate also attains its equilibrium value. Hence, the equilibrium point is indeed stable and a viable cosmological model is arrived.

6.3.3 The superﬂuid dark matter and generalised Chaplygin dark energy

Introducing the last term of the equation of state of the dark matter, namely the inverse term for Λ 6= 0, that indicates a Chaplygin gas behaviour, we are dealing with the complete system of eqs. (357). Thus, the analytic calculation 140 the interacting fluids description of dark matter and dark energy

of equilibrium points is impossible. Of course, we may calculate the divergence of the vector ﬁeld in order to address the possibility of dissipative or explosive behaviour in the overall. It is easily proved that, for B = Γ, which is a prerequisite ~ ~ ~ ~ for the existence of an equilibrium point, ∇V < 0, where V = V(x1, x2, z) the vector ﬁeld of the system, composed by the right-hand sides of eqs. (357); this indicates a dissipative behaviour, at least for the values of the parameters used in our case, consequently, an attracting equilibrium point might indeed exist. This equilibrium point can be located only numerically. We are again reminded that for the equilibrium point to exist, B = Γ. Therefore, we choose A = −1 and Λ = 1, with strict positivity seeming necessary only to the latter, and we assign values to B and Γ in the interval [−0.08, 0.2]; we notice from numerical integration of differential equations that solutions below −0.08 diverge, with those higher than 0.2 yielding negative values for the Hubble rate. Using, A = B = 0.02, we may arrive numerically to the point

P1 (0.668519, 0.331481, 0.53153) .(368)

Other values of B and Γ within the interval, yield an equilibrium point very similar to this one, with their stability being the same. Hence, we will use this case as indicative for the behaviour of the system. Linearising the system in the are of point P1, we obtain the following   dξ1      dN  −1.2387 1.03918 −2.16062 ξ1  dξ2        =  1.2387 −1.03918 2.16062  ξ  ,(369)  dN     2   dξ3 −0.382757 −1.62132 −3.49807 ξ3 dN ∗ where ξi are small linear perturbations around the equilibrium values, xi . The eigenvalues of the system are λ1 = −2.88798 + 1.51784i, λ2 = −2.88798 − 1.51784i and λ3 = 0. Once more, the equilibrium is found to be a non-hyperbolic point, due to the presence of a neutral manifold attribute to the zero eigenvalue.As we mentioned before, however, the Friedmann constraint restricts the phase space to a simple surface, on which lies the equilibrium point, and any trajectory on this surface will be normally lead to P1, since the real part of the other two eigenvalues is negative and indicates stability. This is plainly seen in Fig. 23. Th equilibrium strictly yields we f f = −1 and corresponds to a deSitter expansion of the Universe, with ∼ 67 % of dark energy and ∼ 33 % of dark matter. Thus, it fully agrees with current observations about the late-time accelerating expansion.

It is rather intriguing to see what happens when we get out of the interval [−0.08, 0.2] fro the values of B and Γ, but also when we cease to consider them equal. In fact, these cases sound more realistic, since nature is not necessarily restriced by such “rules of thumb”. 6.3 the classical case (2): working out the model 141

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Figure 23: Intersections of the phase space, for c1 = c2 = 1, wd = 0, A = −1, B = Γ = 0.02 and Λ = 1. The plots correspond to z = 0.53153, x2 = 0.331481 and x1 = 0.668519 respectively (equilibrium P1). Blue streamlines stand for trajectories in the phase space slices, the red spot marks the equilibrium point and the dashed black line denotes the constrain.

Let us ﬁrst consider that A = 1, Λ = 1 and B = Γ = 0.5. The equilibrium point located numerically is

P3 = (0.703536, 0.296464, −0.560748) .(370)

∗ ∗ Notice that it is realistic as long as the x1 and x2 values are concerned, since they both coincide with observational values for the density parameters and fulﬁll the Friedmann constraint; however, the Hubble rate in this equilibrium is found negative, hence a contraction of the 3-d space is implied, that does not match 142 the interacting fluids description of dark matter and dark energy

neither observations nor the theoretical results of the FLRW model. Linearising the system in the area of P3, we have   dξ1      dN  −3.26422 1.89805 1.97606 ξ1  dξ2        =  3.26422 −1.89805 −1.97606 ξ  ,(371)  dN     2   dξ3 4.28266 2.30987 −4.4838 ξ3 dN

that has eigenvalues λ1 = −6.82639, λ2 = −2.81968 and λ3 = 0, thus it presents the same stability issues as the previous two cases; the Hartman-Grobman theorem does not apply, but the fulﬁllment of the Friedmann constraint implies a surface in the phase space where the stability of the equilibrium point is ensured by the other two negative eigenvalues. This is again demonstrated in Fig. 24.

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z 0.0

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Figure 24: Intersections of the phase space, for c1 = c2 = 1, wd = 0, A = 1, B = Γ = 0.5 and Λ = 1. The plots correspond to z = −0.560748, x2 = 0.296464 and x1 = 0.703536 respectively (equilibrium P3). Blue streamlines stand for trajectories in the phase space slices, the red spot marks the equilibrium point and the dashed black line denotes the constrain. 6.4 the loop quantum cosmology case (1): setting up the model 143

Lastly, if we attempt to consider B 6= Γ, no equilibrium point is traced. In fact, when B 6= Γ, the following is observed through numerical integrations of the system (357): the Hubble rate approaches zero, which indicates a static Einstein Universe, but is also a pole of the complete system. As a result, the right-hand- sides of the differential equations tend to inﬁnity as z tends to zero, and the overall behaviour of the system turns explosive. This can also be demonstrated by the divergence of the vector ﬁeld that is strictly positive in this case.

6.4 the loop quantum cosmology case (1): setting up the model

Moving away from the classical limit, we may consider corrections to our initial model, by introducing the Raychaudhuri and Friedmann equations derived for the Loop Quantum theory (see refs. [94, 157, 353] for reviews on Loop Quantum Gravity and refs. [12, 30–33, 382] as insights to Loop Quantum Cosmology). It has been demonstrated that the Friedmann-Lemaître models are viable in the context of the Loop Quantum Cosmology [33, 382], with observational results supporting their viability [177, 398]. The latter has also been proved in many theoretical models, including simple f (R)-corrections [22, 312, 323], f (G)-corrections [221, 323], f (T )-corrections [43, 175, 323] and scalar ﬁelds [223]; recent advances, such as refs. [24, 34, 59, 66, 74, 105, 235, 329] indicate a possibility of early- and late- time accelerating expansions emerging as attractors in Loop Quantum theory. Following from that, we shall assume the Hamiltonian for Loop Quantum Cos- mology 2V H = − ijk ( ) ( ) −1( ){ −1( ) } + 3 3 ∑ ∑ ∑ ε Tr hi λ hj λ hi λ hk λ , V κρV ,(372) γ λ i j k

λσ −i j where hj(λ) = e 2 the holonomies, σj the Pauli matrices, γ = 0.2375 the s √ 3 Barbero-Immirzi parameters, λ = γ a length parameters, V = a3 the space- 2 time volume and ρ = ρdm + ρde the mass-energy density of the Universe. Given some dynamical variable β for which the Poisson bracket with the volume reads γ {β, V} = , 2 the Hamiltonian is simpliﬁed as

sin2(λβ) H = V κρ − 3 ,(373) γ2λ2 and taking into account the Hamiltonian constraint for gravity, H = 0, we may obtain the Loop Quantum theory version of the Firedmann equation

sin2(λβ) κ = ρ .(374) γ2λ2 3 144 the interacting fluids description of dark matter and dark energy

The evolution of the space-time volume, V, over time is given in the Hamilto- nian formalism, by the respective Poisson bracket and reads γ ∂H V˙ = {V, H} = − ;(375) 2 ∂β from this, we obtain the Hubble rate in Loop Quantum Cosmology sin(λβ) H = , γλ so that the dynamical variable becomes arcsin(2λγH) beta = . 2λ Eventually, the Friedmann equation, eq. (374), is rewritten as κ ρ H2 = ρ 1 − ,(376) 3 ρcrit

where ρcrit the critical mass-energy density of the Universe. In a similar manner, utilising the second derivative of the space-time volume over time, we arrive to the Loop Quantum theory version of the Landau-Raychaudhuri equation, κ ρ H˙ = − ρ + P 1 − 2 .(377) 2 ρcrit Notably, in the limit ρ → ∞, the classical Friedmann and Raychaudhuri equations are restored. We shall proceed by investigating the phase space of the two interacting ﬂuids, the superﬂuid dark matter and the generalised Chaplygin gas dark energy, in the context of the Loop Quantum Cosmology, seeking to locate late-time deSitter attractors and check their viability with respect to the classical case. In order to obtain a similar autonomous dynamical system to the classical case, we shall assume the dimensionless phase space variables κρ κρ H2 = = = x1 2 , x2 2 and z .(378) 3H 3H κρcrit Again, our calculations shall be conducted in the domain of the e-foldings number, N. The transformation of derivatives from the cosmic time to the e-foldings number is given in eq. (245) in chapter 4. The evolution of the phase space variables with respect to the e-foldings number, N, is given as dx κ κ 1 = ρ˙ − H˙ , dN 3H de 3H2 dx κ κ 2 = ρ ˙ − H˙ and dN 3H dm 3H2 dz 2 = H˙ . dN κρcrit 6.4 the loop quantum cosmology case (1): setting up the model 145

Substituting from eqs. (351), (352) and (377) and after simple mathematical manipulations, we obtain the following system dx 1 = − 18A2x3z ln2 x + 108ABx4z2 ln x + 36Aw x3z ln x − 324AΓx2x3z3 ln x − dN 1 1 1 1 d 1 1 1 2 1 4AΛx ln x − 36Ax2x z ln x + 3Ax2 ln x + 1 1 − 3Ax ln x − 162B2x5z3 − 108Bw x4z2+ 1 2 1 1 1 z 1 1 1 d 1 3 3 4 3 2 3 2 2 2 3 + 972BΓx1x2z + 108Bx1x2z − 9Bx1z − 12BΛx1 + 9Bx1z − (c1x2 + c2x1) − 18wdx1z+ 4Λw x + 324Γw x2x3z3 + 36w x2x z − 3w x2 − d 1 + w x − 1458Γ2x x6z5 − 324Γx x4z3+ d 1 2 d 1 2 d 1 z d 1 1 2 1 2 2Λ2 Λ 4Λx Λ + 3 2 − 2 + − + + 3 + 2 − 27Γx1x2z 18x1x2z 3x1x2 3 2 36ΓΛx2z 2 , 9x1z 3x1z z 3z dx 2 = − 18A2x2x z ln2 x + 108ABx3x z2 ln x + 36Aw x2x z ln x − 324AΓx x4z3 ln x − dN 1 2 1 1 2 1 d 1 2 1 1 2 1 4AΛx ln x − 36Ax x2z ln x + 2 1 + 3Ax x ln x − 162B2x4x z3 − 108Bw x3x z2+ 1 2 1 z 1 2 1 1 2 d 1 2 2 4 4 2 2 2 2 2 2 + 972BΓx1x2z + 108Bx1x2z − 9Bx1x2z − 12BΛx1x2 + 4(c1x2 + c2x1) − 18wdx1x2z+ 4Λw x 2Λ2x 36ΓΛx4z + 324Γw x x4z3 + 36w x x2z − 3w x x − d 2 − 2 + 2 + d 1 2 d 1 2 d 1 2 2 3 z 9x1z x1 4Λx2 Λx + 2 − 2 − 2 7 5 − 5 3 + 4 2− 2 1458Γ x2z 324Γx2z 27Γx2z x1z 3x1z 3 2 3 2 − 27Γx2z − 18x2z + 3x2 − 3x2 and dz =18A2x2z2 ln2 x − 108ABx3z3 ln x − 36Aw x2z2 ln x − 4AΛ ln x + dN 1 1 1 1 d 1 1 1 3 4 2 2 4 4 3 3 + 324AΓx1x2z ln x1 + 36Ax1x2z ln x1 − 3Ax1z ln x1 + 162B x1z + 108Bwdx1z − 2 3 5 2 3 2 2 2 2 2 − 972BΓx1x2z − 108Bx1x2z + 9Bx1z + 12BΛx1z + 18wdx1z + 4Λwd− 2Λ2 36ΓΛx3z2 4Λx Λ − 324Γw x x3z4 − 36w x x z2 + 3w x z + − 2 − 2 + + d 1 2 d 1 2 d 1 2 2 9x1z x1 x1 3x1z 2 6 6 4 4 3 3 2 2 + 1458Γ x2z + 324Γx2z − 27Γx2z + 18x2z − 3x2z . (379) In the same manner, the effective equation of state from eq. (353) becomes 9Ax2z2 ln x − 27Bx3z3 − Λ − 9(w + 1)x2z2 + 81Γx x3z4 = 1 1 1 d 1 1 2 we f f 2 .(380) 9x1z (x1 + x2) We also know that the system of eqs. (379) is subject to a constraint, arising from the fact that the total matter and energy of the Universe is included and no spatial curvature is considered. Using the Friedmann equation from eq. (376) to express this totality, we may reach following constraint

2 2 x1 + x2 − z x1 + x2 = 1 . (381) That is the Friedmann constraint, proposing that all viable solutions of system (379) must fulﬁll this if the Universe is to be ﬂat and not containing any other 146 the interacting fluids description of dark matter and dark energy

matter field. We should notice that for z < 0 (contracting Universe), the constraint cannot be fulfilled, while for z = 0 (static Universe), it reduces to its classical counterpart (eq. (359); thus, if the constraint is to be fulfilled and deviations from the classical case to be observed, the Hubble rate must be strictly positive and the Universe to expand (z > 0).

6.5 the loop quantum cosmology case (2): working out the model

Little to no analytical work can be carried out as long as the Loop Quantum Cosmology case is examined. Due to the great complexity and nonlinearity of the system (379), no equilibrium points can be calculated analytically; the divergence of its vector field also presents too many non-linear terms -namely logarithms and square roots- as to express analytically some possible attractors and the overall -explosive or dissipative- behaviour of the system. As a result, we shall proceed as before, by numerically examining the phase space of the system in specific cases. We shall assume that c1 = −c2, and more specifically that c1 = −1 and c2 = 1, since numerical solutions of the system indicate opposite signs for these two parameters. wd will be left as a free parameter, since it alters the values of we f f -in this way, we can screen out the effects of the dark energy fluid on the effective equation of state and identify other factors moving it away from esteemed value, we f f = −1. A, B, Γ and Λ will change accordingly to the case we need to examine; we usually assume that B = Γ as in the classical case, though no specific reason for this emerges from qualitative analysis.

6.5.1 The superﬂuid dark matter

In this case, we consider A = Γ = Λ = 0 and B = 1.3 Here the system simpli- ﬁes quite enough fro analytical calculation of equilibrium points; speciﬁcally, the system becomes dx 1 = − 18w2x3z + 3w x2 108x3z3 + 12x z − 1 + dN d 1 d 1 2 2 6 5 4 3 3 2 2 + x1 3wd − 1458x2z − 324x2z + 27x2z − 18x2z + 3x2 − 1 + x2 ,

dx2 2 2 3 3 = − 18w x x2z + x1 3wdx2 108x z + 12x2z − 1 + 4 − dN d 1 2 (382) 6 5 4 3 3 2 2 − x2 1458x2z + 324x2z − 27x2z + 9x2z(3z + 2) − 3x2 + 7 and dz =6w2x2z + w x −108x3z3 − 12x z + 1 + dN d 1 d 1 2 2 5 5 3 3 2 2 + x2 486x2z + 108x2z − 9x2z + 6x2z − 1 . Solving analytically, we obtain three equilibrium points, all of whom yield z = 0, which eventually means that a static Universe is realised; as stated, in this case,

3 B = −1 yields similar results. 6.5 the loop quantum cosmology case (2): working out the model 147 we have a constraint similar to the classical case. From the latter, it is very easy to ∗ ∗ ensure that at least one equilibrium point will be viable, so that x1 + x2 = 1, and obtain we f f = −1. Utilizing this, the three equilibrium points become 1 1 8 32 P (0, 0, 0) , P , , 0 and P − , , 0 .(383) 1 1 2 2 3 9 9 It is clear that the first and the third equilibria have no physical meaning, since they do not fulfill the constraint and, as long the P3 is concerned, negative energy density is implied for the dark energy. As for P2, it fulfills the constraint but also yields we f f = 0, which eventually mean that the respective model is a static Einstein universe filled with dust. Of course, this is mathematically consistent but has nothing to do with the late-time accelerating era we are trying to achieve. Linearising the system around P1, produces the following,   dξ1      dN  −4 1 0 ξ1  dξ2        =  4 −7 0 ξ  ,(384)  dN     2   dξ3 0 0 0 ξ3 dN ∗ where ξi are small linear perturbations around the equilibrium values, xi . The eigenvalues of the system are λ1 = −8, λ2 = −3 and λ3 = 0; the point is proved to be non-hyperbolic and lies outside the constraint, as a result the Hartman- Grobman theorem does not apply. Interestingly, though, at least one stable manifold must exist around the equilibrium, as numerical integrations of the system prove this unviable case to be an attractor (see also Fig. 25). On the other hand, linearisation around P2 reaches the following,   dξ1     1 5  dN  2 2 −9 ξ1  dξ2        =  11 − 5 −9 ξ  ,(385)  dN   2 2   2   dξ3 0 0 −3 ξ3 dN with eigenvalues λ1 = −5, λ2 = −3 and λ3 = 3. Here the Hartman-Grobman theorem does indeed apply and two stable and one unstable manifolds are clearly depicted; it is also very intriguing that one stable and one unstable manifold are vertical on each other and lie on the x1 − x2 plane, deeming P2 to be a saddle. Indeed, from Fig. 25, we can see that solutions are attracted to equilibrium P2 11 in direction x = − x and are repelled by it across the streamline x = x , 2 5 1 2 1 towards equilibrium P1. Another rather intriguing issue we may observe here is that initial conditions on the constraint, x1 + x2 − 1 = 0, do not converge on the equilibrium point P2; Universes initiating with more dark matter than dark energy (x2 > x1) will be immediately attracted to equilibrium point P1, while those initiating with more 148 the interacting fluids description of dark matter and dark energy

1.0

0.8

0.6 2 x

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 25: Intersection of the phase space, for c1 = −c2 = −1, wd = −1, A = Γ = Λ0, and B = 1, corresponds to z = 0 (equilibria P1 and P2). Blue streamlines stand for trajectories in the phase space slices, red spots mark the equilibrium points and the dashed black line denotes the constrain.

dark energy than dark matter (x1 > x2) will be repelled to inﬁnite values of both x1 and x2, that is, towards a Type III or “Big Freeze” singularity. Let us now assume that B 6= 0, so that dark energy also presents elements of complexity; we shall assume that B = Γ = 1. It is proved that the overall structure and behaviour of the system does not change at all; the same equilibrium points exist with the same stability properties, the same unviable attractor P1 and the same “Big Freeze” possibilities appear.

6.5.2 The superﬂuid dark matter and generalised Chaplygin dark energy

Proceeding, one should consider the full equation of state for dark energy, assuming that A 6= 0 and Λ 6= 0 and allowing for Chaplygin elements into the picture. In this case, we move indeed away from the classical-like and wd plays no more an important role. Here, careful choice of wd is again the only way of ensuring at least one viable equilibrium point, although calculations are not as easy as in the previous case. Let us, ﬁrst assume that A = Λ = 1 and B = Γ = 0, so that only the Chaplygin 1 gas elements of the dark energy play a role. In this case w ' − allows for the d 10 existence of one equilibrium point,

P1(1.63532, 0.934467, 0.237806) ,(386) 6.6 conclusions 149

that fulﬁlls the constraint. This equilibrium yields we f f = −0.727273, implying accelerating expansion but not canonical dark energy. Linearisation in the area of it, allows for   dξ1      dN  −19.3934 −3.90595 −94.3442 ξ1  dξ2        =  −2.5222 −9.8034 −36.589  ξ  ,(387)  dN     2   dξ3 1.65979 0.713419 9.3113 ξ3 dN

with eigenvalues λ1 = −11.9581, λ2 = −8.36864 and λ3 = 0.441213. According to the Hartman-Grobman theorem, this point is a saddle containing two stable and one unstable manifolds. As demonstrated in Fig. 26, the unstable behaviour is accompanied by a growing Hubble rate, indicating the existence of a Type I or “Big Rip” singularity, whereas other special types of singularities may also appear (for example, a → ∞, when ρ → 0 and |P| → 0). The case is similar when A = −1 and Λ = 1, with wd ' −1.1 and two unstable manifolds instead of one. Again, a “Big Rip” appears as the Hubble rate grows indeﬁnitely. If A = 1 and Λ = 1, then no equilibrium point can be traced. If one includes B = Γ = 0.01 or B = Γ = −0.01 small deviations may appear. In general, x1, x2 > 1 and 0 < z < 1, so that viability is ensured in terms of the Friedmann constraint, while we f f is generally non-equal to unity, indicating phantom dark energy. A unstable manifold is usually present, deeming the equilibrium points as saddles and driving the system towards some type of singularity, namely the “Big Rip”, since the Hubble rate is positive and growing.

6.6 conclusions

The exploration of the phase space of the two interacting fluid, superfluid dark matter and generalised Chaplygin dark energy, proved fruitful as for the greatest understanding of the existence of stable deSitter attractors in the classical case. Our analysis proved that such attractors do indeed exist and their stability is guaranteed, as long as some conditions hold for the parameters of the system, namely for the superfluidity and the Chaplygin magnitudes. Numerical integrations of the system (357) indicate that Λ must be strictly positive for the stability of the equilibrium to be ensured, while B must always equal Γ, so that an equilibrium exists; furthermore, in order for this equilibrium to be a deSitter expansion and not an anti-deSitter contraction, B and Γ must not exceed specific values, close to 0.2, neither to drop below 0.08 since the stability of the equilibrium is altered. In general, the density parameters x1 and x2 approach values very similar to the observations obtained by the Planck collaboration [9, 13], deeming the equilibria we traced as both stable and viable. Unfortunately, this case does not hold for the entire parameter space, as -for example- the absence of the logarithmic and the Chaplygin terms from the dark energy equation of state might destroy the 150 the interacting fluids description of dark matter and dark energy

1.0

1.4

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1.0 z x

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x1 x1

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0.6 0.8 1.0 1.2 1.4

x2 1 Figure 26: Intersections of the phase space, for c = −c = −1, w = , A = 1, B = 1 2 d 10 Γ = 0 and Λ = 1. The plots correspond to z = 0.237806, x2 = 0.934467 and x1 = 1.63532 respectively (equilibrium P1). Blue streamlines stand for trajectories in the phase space slices and the red spot marks the equilibrium point.

stability as long as the Hubble rate is concerned -and though it, for the density parameters as well. The complete case, however, is stable as long as the afore- mentioned conditions are true. This picture of the classical era is not preserved if one steps into the quantum era and the Loop Quantum Cosmology reflection of the same system. Although the description of the interacting fluids in LQC is capable of producing deSitter and anti-deSitter attractors, those are generally unviable and most of them do not fulfill the Firedmann constraint. Subsequently, the system always at least one unstable manifold and thus present an overall explosive behaviour. Furthermore, the equilibria realised generally admit we f f 6= −1, revealing the emergence of phantom dark energy. In conclusion, the LQC equivalent does not produce viable cosmic evolutions as whole. 6.6 conclusions 151

This last result comes in agreement with ref. [329], where the LQC case may produce stable deSitter attractors under speciﬁc circumstance, but with Type I or III singularities coming to life.

THEPHASESPACEOFTHE k -ESSENCETHEORY 7

7.1 introduction

Among the remaining viable theories that can successfully describe the late-time era is the k-Essence theory, namely the kinetic quintessential dark energy (see refs. [3, 11, 25–28, 38, 72, 106–108, 110, 126, 204, 263, 264, 269, 291, 342, 346, 359] for detailed studies). It is of crucial importance to find a model that can describe in a unified way the dark energy era and the inflationary era. In ref. [291] such a theoretical framework was given in terms of k-Essence f (R) gravity, and it was demonstrated that a viable inflationary era may be generated; however, the results were strongly dependent on the specific model studied, and the true structure of the cosmological solutions must be further revealed. In the following chapter, we provide a detailed study of the phase space of k-Essence f (R) gravity in vacuum, originally presented by [330]. This theory can describe in a viable way the inflationary era too, hence we shall study the phase space in detail, since this investigation may reveal general properties regarding the inflationary attractors. Choosing appropriately the dimensionless variables corresponding to the cosmological system, we constructed an autonomous dynamical system, and found the fixed points of the system and their respective stability by means of simple linear perturbations. We focus on quasi-de Sitter attractors, but also to radiation and matter domination attractors, and study their stability. As we demonstrate, the phase space is mathematically rich since it contains both stable and unstable manifolds. With regard to the inflationary attractors, these exist and become asymptotically unstable, a feature which we interpret as a strong hint that the theory has an inherent mechanism for graceful exit from inflation. The underlying mathematical structures that control the instability of the inflationary attractors are rigorously studied, and the same problem for the radiation-dominated and the matter-dominated eras is also addressed, with the respective attractors being explored. The whole study is performed for both canonical and phantom scalar fields and, as we demonstrate, the canonical scalar k-Essence theory is structurally more appealing in comparison to the phantom one, a result not new at all, since it has been demonstrated in the related literature on k-Essence f (R) gravity. In addition, we find quite intriguing substructures in the phase space, of lower dimension in comparison to the original phase space. These substructures control eventually the stability of the dynamical system, these are the origin of stability. Finally, we also examine in brief the case that no scalar kinetic term is included, and similar results are found.

153 154 the phase space of the k-essence theory

7.2 setting up the model

The k-Essence f (R) gravity theoretical framework belongs to the general f˜(R, φ, X) theory which has the following gravitational action, Z 4 p S = d x −g f˜(R, φ, X) + Lmatter ,(388)

µν √ µν where g is the metric and −g its determinant. Also R = g Rµν is the Ricci 1 scalar and X = ∂ φ∂µφ is the kinetic term of the scalar field. Finally L 2 µ matter stands for the Lagrangian density of the matter fields. In our case we shall assume later on that no matter fields are present, so we will consider the vacuum case Lmatter = 0. For the k-Essence model we shall consider, the generalized function f˜ in the action has the following form,

1 f˜(R, φ, X) = f (R) + c X + G X2 ;(389) 2κ2 1 1 this case leads to a specific category of k-Essence models, to which we shall refer to as “Model I” hereafter. Notice that depending on whether c1 = 1 or c1 = −1, Model I describes a phantom scalar field or a canonical field respectively.

7.2.1 Equations of Motion of the k-Essence f (R) Gravity Theory

Regardless of the specific form of f˜(R, φ, X) and the coupling (or non-coupling) of the kinetic X term to the Ricci curvature, the equations of motion of the theory are derived, as usually, by varying the gravitational action of eq. (388) with respect to the metric tensor, gµν and to the scalar field, φ. The former case yields the field equations for the geometry of the spacetime, that is the generalized Einstein field equations, while the latter yields the evolution of the scalar field. We consider a flat Friedmann-Robertson-Walker (FRW) metric with line element of the form of a˙ eq. (105), where a(t) is the scale factor, and thus H(t) = is the Hubble rate. We a also assume that no matter fields are present in the space-time, so Lmatter = 0. Varying the gravitational action of eq. (388) with respect to the metric tensor, we obtain,

1 κ2 − f˜(R) − RF˜(R) − f˜ φ˙ 2 − 3HF˜˙(R) = 3H2F˜(R) , 2 2 X ∂ f˜ ∂ f˜ where F˜(R) = and f˜ = . As a result, since, ∂R X ∂X 1 R F˜¨(R) = − f˜(R) + F˜(R) − H2 − 2HF˜˙(R) , 2 2 7.2 setting up the model 155 the ﬁeld equations for the f (R) − X theory become κ2 F˜¨(R) − HF˜˙(R) − f˜ φ˙ 2 = 0 . (390) 2 X Varying the gravitational action with respect to the scalar ﬁeld, we obtain,

1 · a3 f˜ φ˙ + f˜ = 0 , a3 X φ ∂ f˜ where f˜ = . Hence, the equation of motion for the scalar field is, φ ∂φ ˙ f˜X φ¨ + 3Hφ˙ + f˜Xφ˙ + f˜φ = 0 . (391) We can now specify the equations of motion for Model I given in Eq. (389). 1 Since f˜ (R, φ, X) = f (R) + c X + G X2, we have, I 2κ2 1 1 ∂ f˜ 1 ∂ f˜ ∂ f˜ F˜ (R) = I = F(R) , f˜ = I = 0 and f˜ = I = c + 2G X . I ∂R 2κ2 I,φ ∂φ I,X ∂X 1 1 where c1 and G1 are constants and can be viewed as free parameters for the models. Concerning c1, its sign can indicate the type of scalar field cosmologies used, with c1 = −1 describing a canonical scalar field, while c1 = 1 describing phantom scalar fields, and c1 = 0 denoting the absence of a kinetic term, which is physically unmotivated, though we will examine this case as well. Consequently, the field equation becomes,

4 2 F¨(R) − HF˙(R) − κ φ˙ c1 + 2G1X = 0 , (392) and the evolution of the scalar ﬁeld, c1 + 2G1X φ¨ + 3Hφ˙) + 2G1X˙ φ˙ = 0 . (393) Having the equations of motion at hand, we can introduce several dimensionless dynamical variables, and we shall construct an autonomous dynamical system, the phase space of which we shall extensively study.

7.2.2 The choice of the dynamical variables and their evolution

In order to examine the cosmological implications and behavior of Model I (389), we shall investigate the mathematical structure of its phase space. In order to do so, we need to introduce appropriate dimensionless phase space variables which will constitute an autonomous dynamical system. Taking into account that in a ﬂat FRW spacetime, we have,

R = 6H˙ + 12H2 and R˙ = 24HH˙ + 6H¨ (394) 1 X = − φ˙ 2 and X˙ = −φ˙φ¨ ,(395) 2 156 the phase space of the k-essence theory

we deﬁne the following ﬁve dimensionless phase space variables,

F˙ f R 1 x = − , x = − , x = , x = κ2φ˙ and x = . 1 HF 2 6H2F 3 6H2 4 5 κ2 H2F (396) The first three of these variables are typical and have been defined as such in many similar f (R) gravity phase space studies [316], however the variables x4 and x5, are needed only in the k-Essence f (R) gravity case. Their evolution will be studied by using the e-foldings number, N, defined as follows,

Z t f in N = H(t)dt ,(397) tin

where tin and t f in the initial and final time instances. The derivatives in respect to the e-foldings number are derived from the derivatives with respect to time, by using, d 1 d d2 1 d2 H˙ d = , = − . dN H dt dN2 H dt2 H dt The equations governing the evolution of the five variables with respect to the e-foldings number are given from the equations of motion, expressed in terms of these variables. Specifically, the evolution of x1 with respect to the e-foldings number is given as, ! dx 1 1 F¨ H˙ F˙ F˙ 2 1 = x˙ = − − − ,(398) dN H 1 H2 F H F F

where, from the ﬁeld equation (eq. (392)), we derived,

2 2 2 F¨ F˙ κ φ˙ c − fDx − = − − c − G φ˙ 2 = x − 1 4 x2x , H2F HF 2H2F 1 1 1 2 4 5 G where f = 1 so that it becomes dimensionless, and also we used the deﬁnition D κ4 of the variables,

H˙ F˙ F˙ 2 = −x (x − 2) , = −x2 . H3 F 1 3 HF 1

Consequently, the differential equation describing the evolution of x1 is equal to,

2 dx c − fDx 1 = −4 + 3x + x2 − x x + 2x − 1 4 x2x .(399) dN 1 1 1 3 3 2 4 5

The evolution of x2 with respect to the e-foldings number is given as,

dx 1 f f˙ H˙ H˙ 2 = x˙ = − + + ,(400) dN H 2 6H3F f H H 7.2 setting up the model 157 where, we used the deﬁnition of the phase space variables,

f˙ R˙ − = − = −4(x − 2) − m , 6H3F 6H3 3 f H˙ f F˙ = −2x (x − 2) , = −x x , 6H2F H2 2 3 6H2F HF 1 2 H¨ where m = − is a dynamical variable of crucial importance. In our study, H3 the dynamical system we shall derive will be autonomous only in the case that the variable m is constant. Thus, the resulting differential equation describing the evolution of the variable x2 is the following,

dx 2 = 8 − m + 4x − 4x + x x − 2x x .(401) dN 2 3 1 2 2 3

Accordingly, the evolution of x3 with respect to the e-foldings number is given as follows, dx 1 R R˙ H˙ 3 = x˙ = − ,(402) dN H 3 6H3 R H From the deﬁnition of the phase space variables, we obtain,

R˙ H˙ H¨ R H˙ = 4 + = 4(x − 2) − m and = x (x − 2) . 6H3 H2 H3 3 6H2 H2 3 3 Consequently, the third differential equation is,

dx 3 = −8 − m + 8x − 2x2 .(403) dN 3 3 We should notice that this differential equation is independent from the rest, since the evolution of x3 depends only on the variable itself and the parameter m. We will show later that this differential equation can be solved analytically for constant m. The evolution of x4 depends strongly on the value of the parameter c1, in such a way that the differential equation governing the evolution has a completely different form when c1 = 0. The evolution is given by,

dx 1 κ2φ¨ 4 = x˙ = .(404) dN H 4 H The second temporal derivative of the scalar ﬁeld φ is derived from the equation of motion for the respective ﬁeld, namely eq. (393), whose form changes drasti- 1 cally with c = 0. To demonstrate this, we substitute X = − φ˙ 2 and X˙ = −φ˙φ¨ in 1 2 eq. (393) and solve it with respect to φ¨, obtaining,

c − f κ4φ˙ 2 ¨ = − ˙ 1 D φ 3Hφ 4 2 .(405) c1 − 3 fDκ φ˙ 158 the phase space of the k-essence theory

When c1 = 0, this expression is simpliﬁed to φ¨ = −Hφ˙ .(406)

In effect, we should consider two distinct forms of Model I, hereafter called “Model Iα” and “Model Iβ”, with the ﬁrst describing the case c1 6= 0 and the second describing the case c1 = 0. The speciﬁc forms of the fourth differential equation are given below:

1 For c1 6= 0, the differential equation describing the evolution of the phase space variable x4 becomes, dx 3 4 = − 3 2 x4 x4 .(407) dN 3 fDx4 − c1 This differential equation resembles a non-linear oscillation, multiplied with a damping term. Similar to Eq. (403), the differential equation (407) is independent from all the other phase space variables and can be analytically integrated, as we will show shortly.

2 For c1 = 0, the differential equation describing the evolution of the phase space variable x4 becomes, dx 4 = −x .(408) dN 4 This differential equation leads to a simple exponential evolution. Obvi- ously, it is also independent and analytically integrated.

Finally, the evolution of the variable x5 is given as, dx 1 1 F˙ H˙ 5 = x˙ = − − 2 .(409) dN H 5 κ2 H3 H F H From the deﬁnition of the phase space variables, we may transform it to dx 5 = 4 − x + 2x x .(410) dN 1 3 5 In conclusion, the dynamical system of the k-Essence f (R) gravity model (389) corresponding to c1 = −1 and c1 = 1 is the following, 2 dx c − fDx 1 = −4 + 3x + x2 − x x + 2x − 1 4 x2x ,(411) dN 1 1 1 3 3 2 4 5 dx 2 = 8 − m + 4x − 4x + x x − 2x x , dN 2 3 1 2 2 3 dx 3 = −8 − m + 8x − 2x2 , dN 3 3 dx 3 4 = − 3 2 x4 x4 , dN 3 fDx4 − c1 dx 1 5 = x˙ = 4 − x + 2x x . dN H 5 1 3 5 7.2 setting up the model 159

while the one corresponding to c1 = 0 is equal to,

2 dx c − fDx 1 = −4 + 3x + x2 − x x + 2x − 1 4 x2x ,(412) dN 1 1 1 3 3 2 4 5 dx 2 = 8 − m + 4x − 4x + x x − 2x x , dN 2 3 1 2 2 3 dx 3 = −8 − m + 8x − 2x2 , dN 3 3 dx 4 = −x , dN 4 dx 1 5 = x˙ = 4 − x + 2x x . dN H 5 1 3 5 In the following sections we shall extensively study the above dynamical systems in detail.

7.2.3 Friedmann Constraint and the Effective Equation of State

Considering all ingredients of the Universe as homogeneous ideal ﬂuids, we may write down an effective equation of state (EoS) as follows,

P w = , e f f ρ where ρ is the energy density of the matter ﬁelds and P is the corresponding isotropic pressure. The effective barotrobic index, we f f , is equal to,

2 H˙ w = −1 − .(413) e f f 3 H2 Given that the Ricci scalar in a FRW space time is R = 6H˙ + 12H2, and, from the R H˙ definitions of the phase space variables, x = = + 2, we have 3 6H2 H2 1 w = − (2x − 1) .(414) e f f 3 3 The effective equation of state must be satisfied by all the fixed points of the dynamical systems (411) and (412), if these fixed points are physical. By looking Eq. (414), it is apparent that x3 determines the value of the EoS parameter we f f in the following way,

1 If the Universe is in a de Sitter expansion phase, so that we f f = −1, then x3 = 2. 1 2 If the Universe is dominated by effective curvature, so that w = − , then e f f 3 x3 = 1. 160 the phase space of the k-essence theory

3 If the Universe is dominated by a pressure-free non-relativistic ﬂuid (dust) so that we f f = 0, which corresponds to the matter-dominated era, then 1 x = . 3 2 1 4 If the Universe is dominated by a relativistic ﬂuids that w = , resulting e f f 3 to the radiation-dominated era, then x3 = 0.

5 If the Universe is dominated by stiff matter so that we f f = 1, then x3 = −1. Another important relation that needs to be fulfilled by the model is the Fried- man constraint, derived from the Friedman equation. Writing down the Friedman equation as, F˙ f R κ2 f φ˙ − − + − X = 1 , (415) HF 6H2F 6H2 2 3H2F where the first three terms correspond to the curvature and the fourth to the scalar field, and by using the definition of the phase space variables, we obtain,

f x + x + x − X x2x = 1 . (416) 1 2 3 6 4 5

Apparently, the constraint depends on the form of f˜I. Since fX = c1 + 2G1X = 2 c1 − fDx4 and c1 may come with two distinct versions of Model I, we have the following two cases:

2 1 When c1 6= 0, so we refer to Model Iα, then fX = c1 − fDx4. The Friedman constraint for Model Iα is the following,

2 c − fDx x + x + x − 1 4 x2x = 1 . (417) 1 2 3 6 4 5

2 2 In the special case of c1 = 0, when we refer to Model Iβ, then fX = − fDx4. Hence, the Friedman constraint for Model Iβ becomes,

f x + x + x + D x4x = 1 . (418) 1 2 3 6 4 5

Having the above at hand, in the next sections we proceed to the analysis of the phase space structure for the k-Essence f (R) gravity models we discussed in the previous sections. As for parameter m, given the fact that it is merely the ratio of the second derivative over the cube of the Hubble rate, it is subject to the specific nature of the spacetime, that is of the specific nature of its matter content. Consequently, the values of m are also very specific, if this is considered to be constant, which is our case. If we consider the three major phases of cosmic evolution, namely the quasi- de Sitter expansion, the matter-dominated era and the radiation-dominated era, we know that the Hubble rate has specific functional forms. Specifically, in the 7.2 setting up the model 161

case of quasi-de Sitter expansion, and H(t) = H0 − Hit, thus m = 0. In the case 2 9 of matter domination, H(t) = , and thus m = − and in the case of radiation 3t 2 1 domination, H(t) = , and thus m = −8. These values of m are the values with 2t major cosmological interest, hence the only values to be used hereafter.

7.2.4 Integrability of the Differential Equations for x3 and x4

As we noted earlier, two of the differential equations composing the dynamical systems (411) and (412), namely eqs. (403) and (407) for Model Iα and eqs. (403) and (408) for Model Iβ, are independent from the other three, meaning that they do not contain any other phase space variables. Consequently, these ﬁrst-order differential equations can be solved independently from the other, and it proves that these can be solved analytically. The independence of these equations, and by extent of the behavior of these two phase space variables, as well as the analytical solutions derived from them, can in fact explain the symmetries we observe later in the corresponding values of x3 and x4 in the equilibria, for all the cases we shall study (for any value of m and fD). They can also explain the stability properties these equilibrium values have, and the corresponding behavior of these variables independently of the others.

3 2 x

0 0 1 2 3 4 5 6 7 N

Figure 27: Analytical solutions derived for the differential equation (403) for different initial values (solid curves correspond to N0 = 0, while thinner dashing cor- 9 responds to greater N0 > 0), for m = 0 (purple curves), m = − 2 (magenta curves) and m = −8 (blue curves). The fast convergence to some equilibrium value is easily observable within 5 to 6 e-foldings.

Beginning with Eq. (403), the analytical solution is √ −2m √ x (N) = 2 − tan −2m(N − N ) ,(419) 3 2 0 where N0 the e-foldings number corresponding to the initial time, i.e. the integration constant of eEq. (397) -usually chosen to be N0 = 0 for simplicity, in the 162 the phase space of the k-essence theory

−44 case of inflationary evolutions, since tin = tPl ' 5.39 10 the initial moment for inflation. Assuming that m =constant, the differential equation (403) has no equilibrium points for m > 0, and has one stable equilibrium point√ for m = 0, ∗ ∗ −2m x3 = 2, and two equilibrium points for m < 0, namely, x3 = 2 ± . Of ∗ ∗ 2 the two, x3 < 2 is the unstable and x3 > 2 is the stable equilibrium point. As a result, the analytical solutions for m = 0 are√x3 = 2 for any N, and for m < 0, the −2m solutions converge rather fast to x∗ = 2 + , with the rate of convergence 3 2 depending on the initial conditions, typically on N0. In fig. 27 we have plotted the behavior of the variable x3 for various values of the parameter m, namely for m = 0 (quasi-de Sitter cosmologies), m = −9/2 (matter-domination cosmology) and m = −8 (radiation-domination cosmology).

1.5 1.5

1.0 1.0

0.5 0.5

4 0.0 4 0.0 x x

-0.5 -0.5

-1.0 -1.0

-1.5 -1.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 N N

Figure 28: Analytical solutions derived for the differential equation (407) for different initial values (solid curves correspond to N0 = 0, while thinner dashing cor- 9 responds to greater N > 0), for m = 0 (purple curves), m = − (magenta 0 2 curves) and m = −8 (blue curves), and for both cases (c1 = −1 on the left, and c1 = 1 on the right). The fast convergence is easily observable within 5 to 6 e-foldings.

As for eq. (408), the solution is,

−N x4 = x4(0)e ,(420)

where x4(0) determined by the initial conditions. This solution converges asymp- ∗ totically, though rapidly to x4 = 0, which is proved to be the equilibrium value for c1 = 0. It is remarkable that this convergence does not depend on any other parameter. Finally, the analytical solution of the eq. (407) is generally derived by means of inverse functions, so it is not possible to present it in closed form. It is however easy to plot it and extract the general behavior, and in Fig. 28 we present the behavior of x4 for various m and c1. What is interesting about it, is that eq. (407) ∗ ∗ ∗ has three equilibrium points, x4 = 0, x4 = −1 and x4 = −1, with the ﬁrst being unstable, and the other two stable for c1 = −1, while all are stable for c1 = 1. Consequently, for c1 = −1 (canonical scalar cosmologies), the solutions ∗ with positive initial values converge rather fast to x4 = 1 and those with negative 7.2 setting up the model 163

∗ initial values converge equally fast to x4 = −1. For c1 = −1 (cosmologies with canonical scalar ﬁelds), the solutions with initial values larger than unity converge ∗ ∗ to x4 = 1, those with initial values smaller than −1 converge to x4 = −1 and, ∗ ﬁnally, those with initial values in the interval [−1, 1] converge to x4 = 0. The rate of the convergence in any of these cases, depends on the choice of fD. In Fig. 28 we present the behavior of the variable x4 for various values of the parameter m and c1.

7.2.5 The three free parameters

One small comment is needed for the free parameters of the two models. Apart from c1 that is inherent from the original theory, two more are included, defined as, H¨ G m = − , and f = 1 .(421) H3 D κ4 Generally, the existence of one free parameter means that the corresponding dynamical system might pass through a number of bifurcations, accordingly to its dependence on this free parameter. The existence of more than one could make the situation far more complicated, with many more bifurcations occurring as different values can be assigned to all the free parameters. This essentially means that the structure of the phase space, and consequently the behavior of the system, may change. The existence of fixed points is questioned, their stability might be altered, attractors can appear and disappear, even chaotic behaviour may arise. Hopefully, in our case things are quite simpler, since our three free parameters are not so arbitrarily chosen actually. Due to the role they play in the model under study, they can be given very specific values. As a result, the parameter space is contained and certain aspects of the bifurcation analysis are similar. More specifically, c1 can take just two values for Model Iα, that are −1 and 1, and only one for Model Iβ, that is 0; the reasons for this have been already explained earlier. The values for the parameter m that will concern as were given earlier, which are m = 0, m = −9/2 and m = −8 corresponding to quasi-de Sitter, matter domination and radiation domination cosmologies respectively. Finally, concerning the third parameter, fD, no value specification is needed beforehand. It will be proved 1 that all values of f > are capable of securing at least one stable manifold in D 3 the phase space, thus ensuring some sort of stability. Given that the Friedmann constraint must be fulfilled, a relation between the other two parameters and fD is given for each of the two distinct models. 164 the phase space of the k-essence theory

7.3 working out the model (1): main analytical and numerical results

Let us begin by examining the phase space of Model Iα, referring to a cosmology with canonical or phantom Cosmological constant. The dynamical system is complosed of eqs. (399, 401, 403, 407 and 410) and it is subjected to constraint (417) and the effective barotropic index of eq. (414); the cases for canonical and quintessential Cosmological constant shall be examined separately, so as to understand better their main differences. Afterwards, the phase space of Model Iβ is presented, referring to a bounding cosmology. Here, the dynamical system is composed of eqs. (399, 401, 403, 408 and 410) and it is subjected to constraint (418), while the effective barotropic index is the same. In the course of this -and the next section- we shall refer to the vector ﬁeld deﬁned by differential equations ~ (399, 401, 403, 407 and 410) or (399, 401, 403, 408 and 410), as F(x1, x2, x3, x4, x5). The qualitative examination of the phase space consists mainly of the location and characterisation of the equilibrium points in the phase space. These points are ~ located by setting F = 0 and solving with respect to the dynamical variables, x1, x2, x3, x4 and x5. In order to characterise the equilibrium points, we shall proceed to the linearisation of each dynamical system around each of the points and solve the linear analogue; according to the Hartman-Grobman theorem, the solutions of the linearised system match the solutions of the actual system very close to the equilibrium point, and subsequently the behaviour of the system close to the equilibrium points is the same as in the linearised system. As a result, we may obtain the eigenvalues and eigenvectors of the linear analogue and thus obtain the corresponding stability of the equilibrium point.

7.3.1 Model Iα (1): the case of canonical dark energy (c1 = −1)

~ Setting F(x1, x2, x3, x4, x5) = 0, we analytically derive sixteen critical points, many of them however come along with complex value in some of their coordinates. This complexity is mainly attributed to the parameter m, out of which we understand that the system is subject to a number of bifurcations, mainly due to the parameter m shifting from positive to negative values. If m > 0, then all the critical points contain at least one complex value, so none of them can be an equilibrium point; this is not really a problem, since m > 0 does not yield physically meaningful solutions. On the other hand, if m = 0, then none of the critical points has a complex value and thus they all may be equilibrium points, however, only six of them exist due to coincidences. These eight equilibria are characterized by high degeneracy due to m = 0 being the transcritical value in this bifurcation, and hence the eight equilibrium points exist in a transitionary state. Finally, if m < 0, then six of the equilibrium points have complex values, hence the remaining ten are equilibrium points. 7.3 working out the model (1): main analytical and numerical results 165

The value of c1 does not play any role in the number of equilibrium points existing, neither does the value of fD. However, either of them may alter the stability of the equilibria, by altering the eigenvalues of the linearized system which is,  dξ1     dN   dξ  ξ1  2      ξ   dN   2  dξ3  = J     ξ3 ,(422)  dN     dξ     4  ξ4   dN ξ  dξ5  5 dN ∗ where {xi } indicate the values of the phase space variables in an equilibrium point and {ξi} denote small linear perturbations of the dynamical variables around them and   ∗ ∗ ∗ ∗ ∗ 2 ∗ 1 ∗ 2 ∗ 2 2x − x + 3 0 2 − x x 2 fDx − c1 x x fDx − c1  1 3 1 4 4 5 2 4 4   ∗ ∗ ∗ ∗   x x − 2x + 4 −2(x + 2) 0 0   2 1 3 2   − ∗  J =  0 0 8 4x3 0 0  ,  ∗ 2 ∗ 2 ∗ 2   3 −3c1x4 + 3 fD x4 + 1 x4 + c1   0 0 0 − 0   − ∗ 22   c1 3 fDx4  ∗ ∗ ∗ ∗ −x5 0 −2x5 0 −x1 − 2x3 + 4 the Jacobian of the system. It is proved, that in order to ensure structural stability for almost every equilibrium point, meaning at least one stable manifold, or at least one eigenvalue with negative real part in the linearised system, fD must have a lower boundary, respective to the c1 value. Since c1 = −1, fD has a lower boundary arising from the ﬁrst eigenvalue of Eqs. (422); this is

1 f > (423) D 3

Furthermore, the values of c1 and fD must also fulfill a specific relationship, in order to secure the viability of the majority of the equilibrium points, as the latter is encoded in the fulfillment of the Friedmann constraint, given in eq. (417), and the effective equation of state, in eq. (414). Beginning from the latter, we may easily rule out any equilibrium point that has a non-matching value for x3. In that case, considering m ≤ 0, only eight equilibrium points are deemed viable, regardless of the values of c1 and fD. Moving onto the fulfillment of the former, we demand that the Friedmann constraint is fulfilled by the eight equilibrium points at any case and for any possible value of c1, m and fD parameters. Thus we derive the following equation √ 1 − x∗ − x∗ − x∗ 3c −2m + 4m = + 1 2 3 = 1√ fD 1 6 ∗ 4 ∗ (424) x4 x5 3 −2m + 4m 166 the phase space of the k-essence theory

3 4

3 2 0 x x

-1

-2

0 -3

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

x1 x1

5 0 x

-1

-2

-3 -2 -1 0 1 2 3

Figure 29: 2-d intersections of the phase space along the x1 direction, for c1 = −1, m = 0, fD = 3 and x4 = −1. Both plots correspond to x3 = 2 (viable cosmological solutions). Blue arrows stand for the vector ﬁeld, green curves for different solutions, black spots for viable equilibrium points and red spots for non-viable equilibrium points.

taking into account the ruling out of two equilibria due to the effective equation of state for m < 0. From eq. (424), we are able to predetermine the value of the last parameter, fD, for the speciﬁc values of the utilized for the other two, c1 and m.

7.3.1.1 Quasi-de Sitter evolution for c1 = −1 and m = 0

Given c1 = −1 and m = 0, Eq. (424) is indeterminate, thus fD is indeed a free parameter. We assume that fD = 1 for any necessary calculation, without any lack of generality. Thus, the six equilibrium points are the following

P1(0, −1, 2, −1, 0) , P2(0, −1, 2, 0, 0) , P3(0, −1, 2, 1, 0) , (425) P4(−1, 0, 2, −1, 0) , P5(−1, 0, 2, 0, 0) and P6(−1, 0, 2, 1, 0) . 7.3 working out the model (1): main analytical and numerical results 167

It is easy to check that all of them fulﬁll we f f = −1 and the Friedmann constraint, thus all six of them are viable as cosmological attractor solutions. Calculating the eigenvalues of the linearized system for these six equilibria, we come to the conclusion that ﬁve out of the six are structurally stable and more importantly asymptotically unstable, due to the presence of both positive and negative eigenvalues, with the sixth being the source of instability. Furthermore, the presence of at least one zero eigenvalue in each of them deems them as irreg- ular and degenerate, a reasonable conclusion due to the transitional value of m in the bifurcation precess (from positive m’s, where no equilibria exist, to negative m’s, where multiple equilibria arise).

4 4

2 2

2 0 2 0 x x

-2 -2

-4 -4

-8 -6 -4 -2 0 2 4 -8 -6 -4 -2 0 2 4

x1 x1

2 0 x

-2

-4

-8 -6 -4 -2 0 2 4

x1 9 Figure 30: Several x -x intersections of the phase space, for c = −1, m = − , f = 3 1 2 1 2 D 1 and x = −1. The ﬁrst two plots correspond to x = (viable cosmological 4 3 2 7 solutions), while the third correspond to x = − (non-viable cosmological 3 2 solution). Blue arrows stand for the vector ﬁeld, green curves for different solutions, black spots for viable equilibrium points and red spots for non-viable equilibrium points. 168 the phase space of the k-essence theory

More speciﬁcally:

1 The P1 and P3 equilibrium points have one stable manifold in the direction of v4 = (0, 0, 0, 1, 0), and one unstable in the direction of v1,2 = (−1, 1, 0, 0, 0); the remaining three are central, depicting a transitionary state.

2 The P2 equilibrium point has two unstable manifolds in the directions v1,2 = (−1, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0); the remaining three are central, depicting a transitionary state.

3 The P4 and P6 equilibrium point have three stable manifolds in the directions v1 = (1, 0, 0, 0, 0), v2 = (0, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and one 1 unstable in the direction of v = , 0, 0, 0, 1 . The remaining one is cen- 1,5 2 tral and depicts a transitionary state. The stability in the directions v1 and v2 is degenerate, due to equal eigenvalues.

4 The P5 equilibrium point has two stable manifolds with degenerate stability, in the directions v1 = (1, 0, 0, 0, 0) and v2 = (0, 1, 0, 0, 0), and two unstable, v4 = (0, 0, 0, 1, 0) and v5 = (0, 0, 0, 0, −1) and the remaining one is central.

In Fig. 29, we present the behaviour of the phase space variables close to the equilibrium points. It is easy to see that the attainment of an equilibrium is usually only along one or two dimensions of the phase space, or depends strongly on the initial conditions. Generally, the system seems to expand along the directions x1 and x5, exponentially leading these variables to infinity (or minus infinity). The presence of asymptotic instability in the dynamical system, after some quasi-de Sitter attractors are reached, is particularly physically appealing. This is due to the fact that inflationary attractors are reached, and then the phase space structure of the k-Essence f (R) gravity reaches some unstable manifolds (certain directions in the phase space), leading to the conclusion that the inflationary attractors become destabilized. This can be viewed as an inherent mechanism for graceful exit from inflation in the k-Essence f (R) gravity theory. Thus combining the present results with those of Ref. [291] which indicated compatibility of k- Essence f (R) gravity theory with the latest Planck data, this makes the theory particularly useful for describing inflationary dynamics.

9 7.3.1.2 Matter-dominated era: The case c = −1 and m = − 1 2 Let us focus in the case of having canonical scalar ﬁelds and matter domination 9 cosmology, which is achieved by choosing c = −1 and m = − . In effect, eq. 1 2 7.3 working out the model (1): main analytical and numerical results 169

(424) yields fD = 3 so that the Friedmann constraint will be satisﬁed at least for those equilibria yielding we f f = 0. The ten equilibrium points in this case read,

1 1 27 7 7 27 P 3, − , , −1, − , P − 3, − , , −1, − , 1 4 2 4 2 4 2 4 1 1 27 7 7 27 P3 3, − , , 1, − , P4 − 3, − , , 1, − , 4 2√ 4 √ 4 2 4 √ √ √ √ 5 + 73 7 + 73 1 5 + 73 7 + 73 1 5 + 73 7 + 73 1 P5 − , , , −1, 0 , P6 − , , , 0, 0 , P7 − , , , 1, 0 , 4√ 4√ 2 4√ 4√ 2 4 √ 4 √2 5 − 73 7 − 73 1 5 − 73 7 − 73 1 5 − 73 7 − 73 1 P − , , , −1, 0 , P − , , , 0, 0 and P − , , , 1, 0 . 8 4 4 2 9 4 4 2 10 4 4 2 (426)

We can easily check that points P2 and P4 yield we f f = −2 and do not satisfy the Friedmann constraint of eq. (417). As a result, they correspond to non-viable cosmologies, however all other equilibrium points yield we f f = 0 and satisfy the Friedmann constraint.

4 4

2 2

3 0 3 0 x x

-2 -2

-4 -4

-6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8

x1 x1

Figure 31: 2-d intersections of the phase space along the x1 direction, for c1 = −1, 9 1 m = − , f = 3 and x = −1. Both plots correspond to x = (viable 2 D 4 3 2 cosmological solutions). Blue arrows stand for the vector ﬁeld, green curves for different solutions, black spots for viable equilibrium points and red spots for non-viable equilibrium points.

In order to account for the stability of the equilibrium points, we calculate the eigenvalues of the linearized system (eq. (422)). Again, all points but one turn to be structurally stable and asymptotically unstable, with at least one unstable manifold. The tenth point is proved unstable. More analytically,

1 Points P1 and P3 have two stable and three unstable manifolds; two of the three unstable manifolds are degenerate due to the equality of the corresponding eigenvalues. 170 the phase space of the k-essence theory

2 Non-viable points P2 and P4 have four stable manifolds and one unstable; two of the four stable manifolds are degenerate as the equality of the corresponding eigenvaluessuggest.

3 Points P5 and P7 have three stable manifolds, in the directions of v1,2 = (−1, 1, 0, 0, 0), v2 = (0, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and two unstable.

4 Point P6 has two stable manifolds, in the directions of v1,2 = (−1, 1, 0, 0, 0) and v2 = (0, 1, 0, 0, 0), and three unstable manifolds.

5 Points P8 and P10 have one stable manifold, in the direction of v4 = (0, 0, 0, 1, 0), and four unstable manifolds.

6 Point P9 has ﬁve unstable manifolds, being an unstable node.

This behavior can partly be seen in ﬁg. 31 where we present the phase space structure in terms of some of the phase space variables.

7.3.1.3 Radiation-dominated Era: The case c1 = −1 and m = −8 Now let us consider the radiation domination cosmologies, in which case m = −8 and let us also investigate the case c1 = −1 which corresponds to canonical scalar 11 ﬁelds. For c = −1 and m = −8, eq. (424) yields f = so that the Friedmann 1 D 5 1 constraint will be satisﬁed at least for those equilibria yielding w = . The ten e f f 3 equilibrium points in this case read,

P1(4, 0, 0, −1, −15) , P2(−4, −2, 4, −1, −15) , P3(4, 0, 0, 1, −15) , P4(−4, −2, 4, 1, −15) ,

P5(−4, 5, 0, −1, 0) , P6(−4, 5, 0, 0, 0) , P7(−4, 5, 0, 1, 0) ,

P8(1, 0, 0, 0, −1, 0) , P9(1, 0, 0, 0, 0, 0) and P10(1, 0, 0, 0, 1, 0) . (427)

7 We can easily check that points P and P yield w = − and do not satisfy the 2 4 e f f 3 Friedmann constraint of eq. (417). As a result, they correspond to non-viable cos- 1 mologies. All other equilibrium points yield w = and satisfy the Friedmann e f f 3 constraint. As for the stability of the equilibrium points, the eigenvalues of the linearized system (eq. (422)) for each of them reveal a complex and unstable nature, similar to the previous case. All points are accompanied by at least one unstable manifold and those providing the greater stability (four stable and one unstable manifolds) are the non-viable two, P2 and P4. More speciﬁcally,

1 Points P1 and P3 have two stable and three unstable manifolds; two of the three unstable manifolds are degenerate due to the equality of the corresponding eigenvalues. 7.3 working out the model (1): main analytical and numerical results 171

2 Points P2 and P4 have four stable and one unstable manifolds; two of the four stable manifolds are degenerate as the equality of the corresponding eigenvalues suggest.

3 Points P5 and P7 have two stable manifolds, in the directions v1,2 = (−1, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and two degenerate unstable manifolds; they also have a central one, depicting a transitionary state.

4 Point P6 has one stable manifold, in the direction v1,2 = (−1, 1, 0, 0, 0) and three unstable, two of which are degenerate since their corresponding eigenvalues are equal; it also has a central manifold corresponding to a zero eigenvalue and depicting a transitionary state.

5 Points P8 and P10 have one stable manifold, in the direction v4 = (0, 0, 0, 1, 0), and four unstable; two of the four unstable are degenerate, due to the equality of the corresponding eigenvalues.

6 Point P9 has ﬁve unstable manifolds, four of whom are pairwise degenerate, being a degenerate unstable node.

In ﬁg. 32 we present the behavior of some phase space variables, for c1 = −1, m = −8, fD = 3. It can be seen that the phase space has attractors, which eventually become destabilized.

7.3.2 Model Iα (2): the case of phantom ﬁelds (c1 = 1)

~ Setting again F(x1, x2, x3, x4, x5) = 0, we derive the same sixteen critical points as in Model Iα. The complex values attributed to the parameter m, appear as well. As a result, previous statements hold in this case as well: if m > 0, then all the critical points contain at least one complex value, so none of them can be an equilibrium point; if m = 0, then none of the critical points has a complex value, with only six of them however to exist due to coincidences; ﬁnally, if m < 0, then six of the equilibrium points have complex values, hence the remaining ten are equilibrium points. The linearized system remains the same, as given by eq. (422). As it was proved, in order to ensure structural stability for almost every equilibrium point, meaning at least one stable manifold, or at least one eigenvalue with negative real part in the linearized system, fD must have a lower boundary, respective to the c1 value. Since c1 = 1 in this case, the lower boundary arising from the ﬁrst eigenvalue becomes 1 f > − (428) D 3

Again, the values of c1 and fD fulfill a specific relationship, in order to secure the viability of the majority of the equilibrium points, as the latter is encoded in the fulfillment of the Friedmann constraint, given in eq. (417), and the effective 172 the phase space of the k-essence theory

6 6

3 2 3 x x

-2

-2 -4

-8 -6 -4 -2 0 2 4 6 -8 -6 -4 -2 0 2 4

x1 x1

-5 5 x

-10

-15

-5 0 5

Figure 32: 2-d intersections of the phase space along the x1 direction, for c1 = −1, m = −8, fD = 3. Blue arrows stand for the vector ﬁeld, green curves for different solutions, black spots for viable equilibrium points and red spots for non-viable equilibrium points.

equation of state, in eq. (414). Demanding that the Friedmann constraint and the effective equation are concurrently fulfilled by all equilibrium points at any case and for any possible value of c1, m and fD parameters, we derive eq. 424 as in the previous case. From this, we should be able to predetermine the value of the last parameter, fD, for the specific values of the utilized for the other two, c1 and m. However, unlike the canonical fields case, c1 = 1 is obviously a pole for fD, so in the cases of phantom scalar fields, fD is considered as a free parameter, chosen as 1 f = for simplicity and without any lack of generality. D 2

7.3.2.1 Quasi-de Sitter evolution with phantom scalar ﬁelds: The case c1 = 1 and m = 0 Let us now consider quasi-de Sitter cosmologies, accompanied by phantom scalar ﬁelds. In this case c1 = 1, the parameter fD is not determined by eq. (424) and 7.3 working out the model (1): main analytical and numerical results 173

1 chosen as f = ; this value is sustained for all phantom scalar ﬁeld cases. The D 2 six equilibrium points of the system when c1 = 1 and m = 0 are,

P1(0, −1, 2, −1, 0) , P2(0, −1, 2, 0, 0) , P3(0, −1, 2, 1, 0) , (429) P4(−1, 0, 2, −1, 0) , P5(−1, 0, 2, 0, 0) and P6(−1, 0, 2, 1, 0) .

The Friedmann constraint of eq. (417) is generally satisﬁed and so is the effective equation of state, yielding we f f = −1, in contrast to the phantom case, where some equilibria where unphysical. Much similarly to the respective phantom scalar case studied earlier, the majority of the manifolds around the equilibrium points are transitionary, since m = 0 is the indicative value for the bifurcation and the turning point for the sign of the eigenvalues. Furthermore, no clear stability is established for any point, with at least one unstable manifold being present in every one. Speciﬁcally,

1 Points P1, P2 and P3 have one stable manifold, in the direction of v4 = (0, 0, 0, 1, 0), and one unstable, in the direction of v1,2 = (−1, 1, 0, 0, 0); the remaining three are central due to the zero corresponding eigenvalues.

2 Points P4, P5 and P6 have three stable manifolds, in the directions v1 = (1, 0, 0, 0, 0) , v2 = (0, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and one unstable manifold; the remaining one is accompanied by a zero eigenvalue, thus being central.

The result is intriguing, since it seems that the canonical k-Essence f (R) gravity theory is more physically appealing in comparison to the phantom scalar k- Essence f (R) gravity. This result is of particular interest since it is aligned with the results of Ref. [291] indicating the same result, that the canonical scalar k-Essence f (R) gravity is compatible with the Planck data, without extreme ﬁne-tuning. In ﬁg. 33 we plot the behavior of several phase space variables for c1 = 1, m = 0, 1 f = . The instability we mentioned above is apparent in all plots. D 2 174 the phase space of the k-essence theory

3 2 3

2 2 0 x x

-2 1

-4

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

x1 x1

5 0 x

-1

-2

-6 -4 -2 0 2 4 6

Figure 33: 2-d intersections of the phase space along the x1 direction, for c1 = 1, m = 0, 1 f = and x = 2 (left and right plot), x = −1. Blue arrows stand for D 2 3 4 the vector ﬁeld, green curves for different solutions and black spots for viable equilibrium points.

9 7.3.2.2 Matter-dominated era with Phantom ﬁelds: c = 1 and m = − 1 2 Let us now consider matter dominated cosmologies with phantom scalar ﬁelds, 9 1 so in this case c = 1, m = − and f = . The corresponding equilibrium points 1 2 D 2 become ten and are the following, 1 1 7 7 1 1 7 7 P1 3, − , , −1, 18 , P2 − 3, − , , −1, 18 , P3 3, − , , 1, 18 , P4 − 3, − , , 1, 18 , 4 2√ √ 4 2 √ 4√2 4 2 5 + 73 7 + 73 1 5 + 73 7 + 73 1 P5 − , , , −1, 0 , P6 − , , , 0, 0 , 4√ 4√ 2 √4 √4 2 5 + 73 7 + 73 1 5 − 73 7 − 73 1 P7 − , , , 1, 0 , P8 − , , , −1, 0 , 4√ 4√ 2 4 √ 4 √2 5 − 73 7 − 73 1 5 − 73 7 − 73 1 P − , , , 0, 0 and P − , , , 1, 0 . 9 4 4 2 10 4 4 2 (430) 7.3 working out the model (1): main analytical and numerical results 175

Of all of them, only ﬁve are viable, satisfying both the Friedmann constraint, eq. (417), and the effective equation of state, eq. (414). Points P1 and P3 satisfy the 7 constraint, but yield w = − , while point P satisﬁes the equation of state, but e f f 3 7 not the Friedman constraint. Finally points P2 and P4 satisfy neither constraint. Concerning their stability, we again derive the eigenvalues of the linearised system of eq. (422), and our analysis indicates that,

1 Points P1 and P3 have two stable and three unstable manifolds; two of the latter are degenerate due to the equality of the corresponding eigenvalues.

2 Points P2 and P4 have four stable and one unstable manifolds; two of the former are degenerate as the equality of the corresponding eigenvalues suggest.

3 Points P5 and P7 have three stable manifolds, in the directions v1,2 = (−1, 1, 0, 0, 0), v2 = (0, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and two unstable.

4 Point P6 has four stable manifolds, in the directions v1,2 = (−1, 1, 0, 0, 0), v2 = (0, 1, 0, 0, 0), v4 = (0, 0, 0, 1, 0) and v5 = (0, 0, 0, 0, 1), and one unstable.

5 Points P8, P9 and P10 have one stable manifold, in the direction v4 = (0, 0, 0, 1, 0), and four unstable manifolds.

The structure of the space is depicted in Fig. 34 for some of the dynamical vari- 9 1 ables, for c = 1, m = − , f = . In this case too, structural instabilities occur in 1 2 D 2 the phase space, and this is a generic feature of the phantom scalar ﬁeld k-Essence gravity.

7.3.2.3 Radiation-dominated Era with Phantom Scalar Fields: The Case c1 = 1 and m = −8 Let us ﬁnally consider the case where phantom scalar ﬁelds are considered for 1 radiation dominated cosmologies, in which case c = 1, m = −8 and f = . Our 1 D 2 analysis indicates that the following ten equilibrium points exist,

P1(4, 0, 0, −1, 32) , P2(−4, −2, 4, −1, 32) , P3(4, 0, 0, 1, 32) , P4(−4, −2, 4, 1, 32) ,

P5(−4, 5, 0, −1, 0) , P6(−4, 5, 0, 0, 0) , P7(−4, 5, 0, 1, 0) ,

P8(1, 0, 0, 0, −1, 0) , P9(1, 0, 0, 0, 0, 0) and P10(1, 0, 0, 0, 1, 0) . (431)

Of these ten, the six satisfy both the Friedmann constraint of eq. (417) and the effective equation of state (eq. (414)), yielding we f f = 0, meaning that the phantom ﬁelds do not affect relativistic matter either. However, though points P1 and P3 1 yield w = − , they do not satisfy the constraint. Finally points P and P do e f f 3 2 4 satisfy neither of the constraints. 176 the phase space of the k-essence theory

5 5

2 0 2 0 x x

-5 -5

-5 0 5 -5 0 5

x1 x1

Figure 34: 2-d intersections of the phase space along the x1 direction, for c1 = 1, m = 9 1 1 − , f = and x = (corresponding to physically viable cosmological 2 D 2 3 2 solutions). Blue arrows stand for the vector ﬁeld, green curves for different solutions, black spots for viable equilibrium points and crimson and red spots for non-viable equilibrium points.

As for their stability, we may again check the eigenvalues of the linearized system of eq. (422), and obtain the following results:

1 Points P1 and P3 have two stable and three unstable manifolds; two of the latter are degenerate since the corresponding eigenvalues are equal.

2 Points P2 and P4 have four stable and one unstable manifolds; of the former, two are degenerate as their equal eigenvalues suggest.

3 Points P5, P6 and P7 have two stable manifolds, in the directions v1,2 = (−1, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and two degenerate unstable manifolds, due to the equality of their eigenvalues; the remaining one is central, as the zero eigenvalue suggest.

4 Points P8, P9 and P10 have one stable manifold, in the direction v4 = (0, 0, 0, 1, 0) and four unstable; two of the latter are proved degenerate due to the equality of the corresponding eigenvalues. 1 The above results can be clearly seen in ﬁg. 35, for c = 1, m = −8, f = . 1 D 2

7.3.3 Model Iβ: the case of bounce (c1 = 0)

We may proceed the analysis with the examination of the phase space of Model Iβ, where c1 = 0. The dynamical system describing such a cosmology is composed of eqs. (399, 401, 403, 408 and 410) and it is subjected to constraint (418) and the effective barotropic index of eq. (414). 7.3 working out the model (1): main analytical and numerical results 177

4 4

2 2

2 0 2 0 x x

-2 -2

-4 -4

-8 -6 -4 -2 0 2 4 6 -8 -6 -4 -2 0 2 4 6

x1 x1

0 x

-2

-4

-8 -6 -4 -2 0 2 4 6

x1 1 Figure 35: Several x -x intersections of the phase space, for c = 1, m = −8 and f = . 1 2 1 D 2 Blue arrows stand for the vector ﬁeld, green curves for different solutions, black spots for viable equilibrium points and crimson and red spots for non- viable equilibrium points.

This system is similar to the previous, with only one differential equation being altered (eq. (408) and some terms being simplified, due to c1 = 0. As a result, many of our previous statements (e.g. the integrability of eq. (403)) are still valid; however, some aspects of the system are different, since this version is much simpler (e.g. the critical points drop from sixteen to four). ~ 1 Setting F(x1, x2, x3, x4, x5) = 0, we analytically derive four critical points, whose coordinates are functions of m. Consequently, the equilibria of the system are subjected to bifurcation, depending on the values of parameter m; more specifically, the four critical points have complex coordinates for m > 0, so none of them is an equilibrium point, and two of them arise with real coordinates for m ≤ 0, so these two are equilibrium points. Both of these equilibria fulfill both the Friedmann constraint (eq. (??) and the effective equation of state (eq. (414)),

~ 1 F(x1, x2, x3, x4, x5) is the vector ﬁeld of the dynamical system, as deﬁned in the previous section. 178 the phase space of the k-essence theory

for each speciﬁc m, thus both equilibria are viable cosmological solutions. Since m > 0 does not correspond to solutions with physical meaning, we shall remain with m ≤ 0 and study the three case with realistic behaviour, m = 0 for the deSit- 9 ter expansion, m = − for the matter domination, and m = −8 for the radiation 2 domination. ∗ The linearised system around a miscellaneous equilibrium point {xi } is given as

 dξ1   4 ∗  dN fDx     2x∗ − x∗ + 3 0 2 − x∗ 2 f x3 ∗x∗ 4 ξ  dξ2   1 3 1 D 4 5  1    2     dN   x∗ x∗ − 2x∗ + 4 −2(x∗ + 2) 0 0  ξ2  dξ3   2 1 3 2      =  ∗  ξ  ,    0 0 8 − 4x 0 0   3  dN   3     dξ4  ξ     0 0 0 −1 0   4  dN      ∗ ∗ ∗ ∗ ξ5 dξ5 −x5 0 −2x5 0 −x1 − 2x3 + 4 dN (432) where ξi the linear perturbations of the state variables, xi around the equilibrium point. Unlike the previous case, the fD parameter is not determined from the value of m via the constraint; thus it can be perceived as a free parameter. Furthermore, the coordinates of the equilibrium points are not depending on fD, and it is removed ∗ ∗ from the linearised system (since x4 = x5 = 0) and by extent to the eigenvalues and eigenvectors of it governing the behaviour close to the equilibrium points. For simplicity, and without any loss of generality, we set fD = 1 for any numerical calculation or plot we are about to conduct. For a quasi-de Sitter evolution, m = 0, so we may derive the following two equilibrium points,

P1(−1, 0, 2, 0, 0) and P2(0, −1, 2, 0, 0) .(433) Using the linearized system from eq. (432), we may reach to the following structure:

1 Point P1 has three degenerate stable manifolds, in the directions v1 = (1, 0, 0, 0, 0), v2 = (0, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and one unstable manifold, in the direction v5 = (0, 0, 0, 0, 1); the remaining manifold, in direction v1,2,3 = (−3, 4, −1, 0, 0) is central , since the corresponding eigenvalue is zero.

2 Point P2 has one stable manifold, in direction v4 = (0, 0, 0, 1, 0), and one unstable manifold, in the direction v1,2 = (−1, 1, 0, 0, 0); the remaining three manifolds are central, since their eigenvalues equal zero. 9 Given m = − , where matter domination is implied, the equilibrium points 2 become, √ √ √ √ 5 + 73 7 + 73 1 5 − 73 7 − 73 1 P − , , , 0, 0 and P − , , , 0, 0 .(434) 1 4 4 2 2 4 4 2 7.4 working out the model (2): additional characteristics 179

From the linearized system from eq. (432), we obtain the eigenvalues of each. Then the stability structure of the ﬁxed points are as follows,

1 Point P1 has three stable manifolds, in the directions v1,2 = (−1, 1, 0, 0, 0), v2 = (0, 1, 0, 0, 0) and v4 = (0, 0, 0, 1, 0), and two unstable manifolds.

2 Point P2 has two stable manifolds, in directions v4 = (0, 0, 0, 1, 0) and v5 = (0, 0, 0, 0, 1), and three unstable manifolds.

Finally, for the radiation-dominated era, given m = −8, the equilibrium points are, P1(−4, 5, 0, 0, 0) and P2(1, 0, 0, 0, 0) .(435) From the linearized system from eq. (432), the stability of each is obtained as follows,

1 Point P1 has two degenerate stable manifolds, in the directions v1,2 = (−1, 1, 0, 0, 0) and v2 = (0, 1, 0, 0, 0), and two degenerate unstable manifolds; the remaining one yields a zero eigenvalue, being a central central and depicts a transitionary state.

2 Point P2 has one stable manifold, in direction v4 = (0, 0, 0, 1, 0), and four unstable manifolds; two of the latter have equal eigenvalues, being degenerate.

Thus in the c1 = 0 case, certainly the phase space contains some stable attractors, and an interesting property of the phase space is analyzed in the next section.

7.4 working out the model (2): additional characteristics

As we stated, despite the degeneracy and the instabilities of the model, several interesting characteristics arise in the phase space for both models, Iα and Iβ. Among these characteristics, we can locate some stable attractors, that can be calculated analytically.

7.4.1 Model Iα: a possible 2 − d attractor

Very important information about a dynamical system arise from the divergence ~ of its vector ﬁeld, F(x1, x2, x3, x4, x5), which in the case of Model Iα, yields

1 c − 3 f 2c (c − 3 f ) ∇~ F~ = 13 + 2x − 9x + − 1 − 1 D + 1 1 D ,(436) 1 3 2 2 2 fD c1 − 3 fDx4 (c1 − 3 fDx4) 1 for c = 1 and f > . Generally, a dynamical system is explosive if ∇~ F~ > 0, 1 D 3 conservative if ∇~ F~ = 0, or dissipative if ∇~ F~ < 0, meaning that supervolumes of initial values are increasing, non-changing, or decreasing over time, respectively. 180 the phase space of the k-essence theory

In our case, the sign of the divergence of the flow changes, which means that the system is neither explosive, neither conservative, nor dissipative, but rather a mixture of all these depending on the area of the phase space where the flow operates on the initial values. According to the Poincaré-Bendixon theorem, the change of sign of the flow of a dynamical system indicates the existence of an attractor or a repeller in the phase, such as a stable or an unstable limit cycle. The case of our dynamical system is similar, since the flow, ∇~ F~ turns zero along a specific three-dimensional curve; this curve is defined as q 3 fD + c1 2 fD(13 + 2x1 − 9x3) − 3 − (3 fD − c1) 3 fD + c1 3 fD − 9 + c1(13 + 2x1 − 9x3) 2 x4 = , 6 fD fD(13 + 2x1 − 9x3) − 1 (437) which is valid only when 9 f x − 13 f + 1 x1 6= D 3 D . 2 fD The curve defined in eq. (437) can be further specified as follows: √ − − + − + − + 2 2 4x1 18x3 23 5 2x1 9x3 1 It becomes x4 = in the case of 6x1 − 27x3 + 36 deSitter expansion ( fD = 1) and canonical Cosmological constant (c1 = −1); the quantity on the right-hand side is generally real and positive for x1 > 1 (9x − 16), so x takes realistic values across this curve. However, this 2 3 4 curve is not proved to be an invariant under the flow of the system, thus it can not be categorised as an attractor. √ √ 2 15 −8x1 + 36x3 − 49 + 6x1 − 27x3 + 33 2 It becomes x4 = − in the case of 9(6x1 − 27x3 + 38) matter domination ( fD = 3) and canonical Cosmological constant (c1 = −1); again, the quantity on the right-hand side is generally positive, so x4 takes 1 realistic values across the curve, for x > (9x − 16). 1 2 3 √ √ 5 19 −88x1 + 396x3 − 533 + 22x1 − 99x3 + 119 2 3 It becomes x4 = − in 33(22x1 − 99x3 + 138) 11 the case of radiation domination f = − and canonical cosmologi- D 5 cal constant (c1 = −1); once more, the quantity on the right-hand side is generally positive, so x4 takes realistic values across the curve, for x1 > 1 (9x − 16). 2 3 √ 2 − 16x1 − 72x3 + 89 + 4x1 − 18x3 + 23 4 It becomes x4 = in the case of phan- 6x1 − 27x3 + 33 1 tom fields (c = 1 and f = ); here, the quantity on the right-hand side is 1 D 2 7.4 working out the model (2): additional characteristics 181

generally negative, so x4 would take non-realistic complex values across the curve. As a result, the attractor cannot exist in the case of phantom ﬁelds.

7.4.2 Model Iβ: a possible 1 − d attractor

As in the previous model, the flow of the system, defined as ∇~ F~ does not maintain its sign when c1 = 0, thus it is impossible to define the system as conservative, dissipative or explosive; more specifically ~ ~ ∇F = 18 + 2x1 − 9x3 ,(438) that changes sign astride a plane supersurface. It is very interesting that the flow of the system does not depend on the variables x2, x4 and x5, hence the phase space is compactified on a 2 − d surface of x1 and x3, the latter of which is analytically determined; furthermore it does not depend on the parameters m and fD, thus the line is not subject to bifurcations as the position and stability of the equilibrum points do. Demanding that ∇~ F~ = 0, we trace this supersurface as

9 x = − (2 − x ) .(439) 1 2 3

This supersurface is fully determined over the e-foldings number, N, since x3 is given as an analytic solution of eq. (403); thus, eq. (439) provides us with an analytic solution for x1 as well, in the form √ 9 2m √ x (N) = −7 − tan 2m(N − N ) .(440) 1 4 0 Solutions of eq. (399) are thus driven by solutions of eq. (403), and by extent they are attracted towards the value √ 9 2m √ x? = − tan 2m .(441) 1 4 This value does not correspond to an equilibrium point, accept for the case of deSitter expansion (m = 0). Given this result, one should also give notice to the strong intermingling of the F(R) function and its rate of change, F˙(R), with the curvature. If we substitute the variables x1 and x3 from their deﬁnitions (eq. (396), to the eq. (439), we obtain the following relation F˙ R = − 9H .(442) F 6H Since R = 6H˙ + 12H2 in a ﬂat Friedmann-Lemaître-Robertson-Walker space-time, the above equation can be rewritten as

F˙ H˙ = − 7H ,(443) F H 182 the phase space of the k-essence theory

-5 1 x

-10

-15

0 1 2 3 4 5 6 7 N

Figure 36: Analytical solutions of eq. (399) that correspond to the 1 − d attractor existing 9 for c = 0. Blue curves stand for m = 0, purple curves for m = − and red 1 2 ones for m = −8; the solid curves denote initial conditions for N0 = 0, while dashing becomes thinner as N0 > 0 grows.

that can be immediately integrated and yield the derivative F(R); if further inte- grate, this allows us to obtain the f (R) function that realises the speciﬁc cosmological model, proposing that the Hubble rate is given with respect of time, and hence with respect to the scalar curvature. If, for example, we consider H(t) = H0 − H1t, where H0 and H1 are constants, that corresponds to the quasi-deSitter expansion, we may easily rewrite eq. (443) as F˙ H1 H1 R = − − 7 −H0 + + + H0 , F H1 R 2 12 −12H0 + 2 + 12 + H0 which is solved analytically and yields

−7 (12H1+R)R −12H1 F(R) = F0e 24 (6H1 + R) ,

where F0 the integration constant. In the same manner, if we consider H(t) = 2 , where w is the barotropic index, that corresponds to any kind of typical 3(1 + w) matter ﬁeld, we rewrite eq. (443) as

F˙ R(1 + w) 7R(1 + w) = − − p p , F 2 2 − 4 − R(1 + w) 3(1 + w) 2 − 4 − R(1 + w)

that yields the following solution, √ (3w+17)(R(1+w)−4)( 4−R(1+w)+3) − 9(1+w)2 F(R) = F0e . 7.4 working out the model (2): additional characteristics 183

7.4.3 Inﬁnities encountered in the models

Before moving onto Model Iβ, we are bound to discuss a couple of issues arising for Model Iα, that lead to mathematically inconsistent or physically non-viable situations.

7.4.3.1 Zero equilibrium values for x5

The most typical valuue of state variable x5 in an equilibrium is zero. Despite the system is generally asymptotically unstable in the direction of x5, the equilibrium ∗ value alone should trouble us. Considering x5 = 0 means, from its definition in eqs. (396), that either F(R)∗, or H∗ should be infinite. The former means that the change of f (R) in a cosmological solution should be infinite, and hence f (R) would increase or decrease infinitely with an infinite rate; the latter means that the Hubble rate of a cosmological solution tends to infinity, and by extent the spacetime expands to infinity or contracts to zero with an infinite rate. Obviously, neither of these can be true. Consequently, any equilibrium point with such a zero x5 value should by treated as unviable, at least as long as there is a stable ∗ manifold leading towards x5 = 0.

7.4.3.2 Inﬁnities for c1 = 1

2 c1 Beginning from eq. (407), we can clearly see the existence of poles for x4 = ± 3 fD and eventually for r c1 x4 = ± .(444) 3 fD

Of course, these poles are purely imaginary for c1 = −1 and hove no much importance in the case of a canonical Cosmological constant. However, assuming 1 c = 1 -and by extent f > - the poles are purely real and appear as two straight 1 D 3 superplanes along the phase space, in the form of r 2 x = ± , 4 3 1 assuming f = as we did in the visualisation of the vector field and the trajec- D 2 tories in the phase space. Close to these superplanes, the derivative of x4 tends to infinity, and by extent the values of x4 change rapidly towards infinity (or minus infinity) as well. This behaviour is not natural and splits the phase space in three discrete and isolated parts, each of whom has a specific sink; any initial values r r 2 2 of x > tends to x∗ = 1 and those with initial values for x < − tend 4 3 4 4 3 r r 2 2 to x∗ = −1, while the initial conditions in − < x < tend to x∗ = 0. 4 3 4 3 4 184 the phase space of the k-essence theory

As a result, when a phantom ﬁeld is present instead of a canonical Cosmological 2 constant, the state variable x4 may tend to zero through a stable mechanism . This indicates that φ˙ = 0, hence that the scalar ﬁeld remains constant over time, is a stable solution for our model.

7.5 conclusions

The two models analysed in this paper are simple but essential cases of an f (R) theory of modified gravity minimally coupled with a Galileon field, in such a way as to yield cosmological solutions with physical meaning and to explain endogenously the specific cosmological eras, such as the initial deSitter expansion (cosmological inflation), the radiation- and matter-dominated phases, and finally the late-time accelerating expansion. They were compactified so as to transform the cosmological dynamics of the theory in a dynamical system of five state variables; the analysis of the phase space of this dynamical system is capable to reveal the fundamental behaviour of the specific models by means of the equilibria and their corresponding stabilities. Proceeding to this analysis, we isolated three major cases, identified as c1 = −1 (the canonical Cosmological constant cosmologies), c1 = 0 (bounce cosmologies) and c1 = 1 (the phantom fields cosmologies). These three cases are further specified by the choice of a specific matter fields content for the Friedmann space-time, 9 namely m = 0 for the simple deSitter expanding space-times, m = − for matter- 2 dominated space-times, and m = −8 for radiation-dominated space-times. These total nine cases can be studied as different static versions of a specific system, which is more or less the way of analysis we used; however, they can be studied as three different systems (concerning the values of c1) that are subject to a bifurcation (concerning the values of m), since the latter is supposed to vary slowly throughout the evolution of a Friedmann -Λ-CDM universe. In this sense, we may observe some interesting results, summarised as follows. First of all, given that c1 = −1 or c1 = 1, the bifurcation of m from positive values to zero creates six equilibrium points, and from zero to negative values to further four; two of these four in the cases of Cosmological constant, and all four in the cases of phantom fields, are unviable equilibria, since they do not fulfill the Friedmann constraint and/or the effective equation of state for the specific matter fields content. Furthermore, the appearance of equilibrium points occurs in such a way that specific symmetries are present and easily observed in the values of x1 and x2 state variables, even more in the values of x4 and x5 state variables. Taking into account the definitions of the state variables, the symmetries observed in x1 and x2 are symmetries concerning the specific form of the f (R) function; on the other side, the symmetries observed in x4 correspond to the behaviour of the scalar field, φ; finally, x5 has a usual equilibrium value at zero, which denotes

2 This was also demonstrated by the analytical solutions of eq. (407) 7.5 conclusions 185

an infinite rate of increase (or decrease) for either the Hubble rate, or the F(R) function 3. The four emerging (and partially unviable) equilibrium points are conceived as two pairs, mirrored on x4 = 0, which denotes the constancy of the scalar field, ∗ since four of their coordinates are exactly the same and the fifth (the x4) takes respectively the values −1 and 1; in fact the stability of two mirrors the stability of the other two, qualitatively (the number of stable manifolds) and quantitatively (the direction of the stable and unstable manifolds). The stability of each isolated equilibrium is generally preserved through the bifurcations, with one important note: moving from m = 0 to m < 0, the number of central manifolds decreases and stable manifolds replace them. The remaining six (always viable) equilibrium points can also be perceived as two groups of three, mirrored on x4 = 0, with all other coordinates being equal (in each group), and x4 taking the vaues −1, 0 and 1. The stability of each group seems correlated, with points with relatively more stable manifolds being grouped together and those with fewer stable manifolds alike; in the same ∗ manner, the equilibria of a group with x4 = 0 are proved to be relatively more unstable that the other two. This peculiar symmetry reveals a fundamental problem ∗ ∗ in the case of φ = const., that repels solutions towards x4 = −1 or x4 = 1. The gradual disappearance of central manifolds as m moves from zero to negative values is observed here as well; following the typical scheme for the evolution of the universe, some of the central manifolds are preserved when m = −8 (radation- 9 dominated era) bu are completely erased for m = − , proving that the values of 2 parameter m are not decreasing linearly and uniformly. Given the degenerate case of c1 = 0, the number of equilibrium points shrinks ∗ ∗ to only two. Here x4 = x5 = 0 always; yet, both equilibria are proved viable according to the Friedmann constraint and the effective equation of state. The two equilibrium points generally preserve their stable manifolds as the values of 9 m move from 0 to − and −8; the first of these points for which x∗ < x∗, has 2 1 2 more stable manifolds. Once more, the transition from m = 0 to m = −8 does not completely transform the central manifolds to stable ones, as it happens when 9 m = −8 changes to m = − ; the standard model for cosmic evolution is somehow 2 present in this phase transition, concerning the stability of the equilibria. Second, a possible attractor appears in all three major cases. For c1 = 0 the attractor is a 2-d supersurface, that connects the x1 and x3 state variables and does not depend on any parameter values. When c1 = −1 or c1 = 1, the attractor is a 3-d supersurface, that connects the x1, x3 and x4 state variables and depends strongly on the choice of fD and c1; however, its complexity is such that its ex-

∗ 3 Almost ironically, the equilibrium points for which x5 6= 0 are usually the unviable points for m < 0. Even more ironically, two of these are always found asymptotically stable in four directions; the ∗ equilibria for which x5 = 0 are usually found to be asymptotically unstable, at least in directions such as x5. 186 the phase space of the k-essence theory

istence is not guaranteed. Interestingly, in any of the above cases the possible attractor connects the rate of change of the F(R) function (from the x1 variable) to the curvature scalar R (from the x3 variable) and to the rate of change of the scalar field (from the x4 variable). However, the two of these variables not only correspond to specific symmetries in the model, as observed in the equilibrium points, but also determine the effective equation of state and the Friedmann constraint respectively; furthermore, their dynamical equations are separated from the other variables in a certain way, leading to their integrability and triviality. Finally, if we observe this separation of eqs. (403) and (407 or 408) from the other three and analyse their integrability, we come to the easy result that both x3 and x4, or with other words the scalar curvature and the rate of change of the scalar field, are trivially and isolatedly evolving towards an equilbirium value. As long as the scalar field is concerned, the attained equilibrium value(s) are normal and correspond to viable cosmological solutions; when the scalar curvature is taken into account, we observe that aside from the case of deSitter evolution, the equilibrium attained does not correspond to the barotropic index of the specific matter fields content, and thus relates to unviable cosmological solutions. As a consequence of the above, the system is characterized by two extremely different states. On the one hand, an asymptotic instability is observed in almost every equilibrium point that was noted. This instability is further amplified, if we take into consideration that the greater stability resolves around non-viable cosmological solutions. Concerning quasi-de Sitter fixed points, this asymptotical instability after an attractor is reached, may be viewed as an inherent mechanism for graceful exit in the k-Essence f (R) gravity theory. On the other hand, a high degeneracy is easy to be noted, given the fact that many equilibrium points emerge, and have specific symmetries in their coordinates, as well as the eigenvalues and eigenvectors of the linear perturbations, that characterize their stability. Also two of the dynamical equations are separated from the rest and integrated, resulting to trivial solutions for x3 and x4. The third state variable, x1, is (in some cases) strongly interconnected with the x3 and x4 via an attractor, and thus its behavior is analytically traced and found diverging from the noted equilibria (viable or not). Finally, many of the stable or unstable manifolds around the equilibrium points are proved degenerate. A possible resolution of this degeneracy would be the transformation of the system into another and the subsequent reduction of its dimensions from five to four, or ever three. This would expel the degeneracy and would give us a clearer picture for the general behavior of the system, under the aforementioned bifurcations. However, the fundamental instabilities of the system are not expected to alter following such a transformation. This issue is more probably an issue of the theoretical framework out of which the models were derived, rather than of the specific dynamical system, indicating for the quasi-de Sitter fixed points that the final attractors are unstable asymptotically and thus this is a strong hint that the theory possesses an internal structure that allows a graceful exit from inflation. Part V

CONCLUSION

CONCLUSION 8

The idea of large scale structure and evolution of the Universe as a whole is at least as old as mankind itself. Initially, it was addressed in the context of philos- ophy or religion, as a mean to explain the appearance and to trace the purpose of man on Earth. Following Sir Isaac Newton and the first formal theory of gravity, the same question attained a scientific status, as the formation ad evolution of the observable Universe could now put under the governance of laws and be investigated as a consequence of these very laws. With the pioneering works of James Jeans, Albert Einstein, Edwin Hubble, Willem deSitter, Alexander Fried- mann, Georges Lemaître, Richard Tolman and others, within the context of New- tonian or relativistic gravity, many possibilities emerged as for the understanding of the Universe in the large scale. However, both Newtonian theory of Gravity and General Relativity contained “flaws”, that is cases where they ceased to apply due to their restricted mathematical formulation or their inelastic axioms. In order for these blank spots to be filled, alternative theories that extended or modified the original theory emerged as well. Some of them were merely phenomenological modifications of General Relativity, while others attempted to account for quantum or string corrections, supposing that a -yet unknown- quantum theory of gravity lied as the secret source of any effective theory. Many of these attempts have been disregarded during the last decades by means of observations. Others remain viable, since they continue to agree with both General Relativity and the current observations. In this dissertation, we dealt with two of these theories, the f (R) and the f (G), along with their extensions, such as the f (R, φ, X) and the f (R, G, φ). We briefly discussed the structure of the theories and presented the field equations and the conservation laws for each of them. Furthermore, we showed what the implications are for such a theory to produce a viable cosmological model and how one such model can be reconstructed and compared to observational evidence from Planck and BICEP2/KeckArray. Our results were encouraging, proving that both f (R) and f (G) theories are viable competitors of General Relativity in explaining the evolution of space-time, especially in the cosmic eras where the latter seemingly fails, namely the early- and late-time accelerating expansions. In the conclusion of the dissertation, we presented two exquisite models of modified gravity, that are analysed in terms of their phase space. We construct two dynamical systems that encapsulate the fundamental issues of each cosmological model and attempt to prove that specific inflationary or deSitter attractors exist. The analysis indicates that both the classical relativistic case and the k-essence f (R) theory reach stable conditions, yet several issues might appear

189 190 conclusion

concerning the values of speciﬁc cosmological parameters. In sharp contrast, the Loop Quantum Cosmology approach fails to produce stability conditions in either canonical or phantom dark energy. Concluding, we should note that none of these can be viewed as a complete and rigorous discussion, since a cosmological model in terms of modiﬁed theories of gravity comes along with great complexity and has to be constantly compared to both observations and classical relativistic modeling, in order for their viability to be proved beyond any shadow of a doubt. BIBLIOGRAPHY

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