Models of Disformally Coupled Dark Energyrepositorio.ul.pt/bitstream/10451/35533/1/ulfc121899_tm... · 2019-03-27 · Models of Disformally Coupled Dark Energy Elsa Maria Campos Teixeira

UNIVERSIDADE DE LISBOA

FACULDADE DE CIÊNCIAS

DEPARTAMENTO DE FÍSICA

Models of Disformally Coupled Dark Energy

Elsa Maria Campos Teixeira

Mestrado em Fı́sica

Especialização em Astrofı́sica e Cosmologia

Dissertação orientada por:

Doutor Nelson Nunes

e Prof. Doutora Ana Nunes

2018

Acknowledgments

I would like to take advantage of this opportunity and express my gratitude to everyone who, directly

or indirectly, had a significant contribution to the conclusion of this dissertation. First, I would like to

thank my supervisors, Doutor Nelson Nunes and Prof. Doutora Ana Nunes, for introducing me to the

world of research, for the unending discussions and mostly, for their enthusiasm and dedication to this

project. I have learned plenty throughout this process and I am very grateful.

I thank everyone in the Cosmology group, for always making me feel welcome and always being

kind. In particular, I thank José Pedro Mimoso and Francisco Lobo for the guidance, the encouragement

and insightful comments. Also, to Inês, Rita and Isma for always being around, for the fruitful discus-

sions and above all, their friendship. To Bruno, for all the unyielding care and for everything he has

taught me.

To my closest friends, for always cheering me on and for all the support over the years. And finally, a

heartfelt thanks to my family, in particular my parents, for their endless support and for always reminding

me of what’s right, and to my cousin, for always inciting my interest in science. I dedicate this work to

my sister, whose patience, optimism and advice were more valuable than she could ever imagine.

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Resumo

A Cosmologia é o estudo do Universo, ou cosmos, como um todo. A Cosmologia padrão surgiu

com a formulação da teoria da Relatividade Geral, em 1915, por Albert Einstein que, desde então, se

tem mostrado bem sucedida ao descrever a natureza do nosso Universo, tanto a pequenas como a largas

escalas. Mostrou-se também capaz de fazer previsões rigorosas que, só mais tarde, com o avanço da

tecnologia, puderam ser confirmadas, como é o caso das recém-detectadas ondas gravitacionais. Por

isso, o modelo padrão da Cosmologia baseia-se essencialmente na teoria da gravitação de Einstein e nas

suas implicações cosmológicas. No entanto, quando em 1998 se descobriu que o nosso Universo está

a experienciar uma fase de expansão acelerada, não parecia haver explicação para este fenómeno. Esta

adição ao nosso conhecimento acerca do Universo levou à necessidade de considerar extensões à teoria

da Gravitação em vigor.

Sabe-se que, todo o tipo de matéria conhecida e detectada até à data, incluı́da no Modelo Padrão da

Fı́sica de Partı́culas, interage gravitacionalmente de forma atractiva (e nunca repulsiva), o que significa

que, tendo apenas isso em conta, o Universo não se deveria estar a expandir. Neste sentido, um dos

maiores desafios da Cosmologia diz respeito à determinação da composição do Universo. Presentemente,

uma das generalizações mais bem aceites consiste em assumir que a expansão acelerada do Universo é

provocada pela presença de uma componente de matéria/energia, desconhecida até à data, caracterizada

por uma pressão eficaz negativa. Na realidade, através do ajuste dos dados observacionais, é possı́vel

inferir que esta fonte desconhecida teria de ser a mais abundante entre todos os constituintes do Universo.

A esta componente dá-se o nome de Energia Escura. Postula-se ainda a existência de um tipo de matéria,

denominada Matéria Escura, necessária para ajustar correctamente os dados observacionais, que sugerem

que deveria existir muito mais matéria do que aquela que se observa na realidade. Esta matéria não parece

emitir nem absorver radiação, pelo que não pode ser detectada por interacção electromagnética.

O próprio Einstein, enquanto procurava uma solução para a sua teoria que permitisse descrever um

Universo estático, verificou que seria necessário incluir uma contribuição proveniente de uma compo-

nente de matéria com pressão negativa. Tratando-se de algo pouco usual, Einstein optou antes por adi-

cionar à sua teoria a famosa constante cosmológica Λ. Mais tarde mostrou-se que esta solução apresenta

instabilidades mas serviu de inspiração às tentativas futuras de explicar a expansão acelerada do Uni-

verso. Assim, o modelo mais simples de energia escura é o da constante cosmológica Λ, que representa

uma fonte cosmológica com pressão pΛ = −ρΛ (onde p representa a pressão e ρ a densidade de energia

que, por questões de consistência, se assume sempre positiva). Com a adição de uma componente de

matéria escura obtém-se o modelo padrão da cosmologia actual: o modelo ΛCDM. Ainda assim, este

modelo enfrenta alguns obstáculos conceptuais e, por isso, considera-se extensões em que a energia es-

cura é descrita através de um campo escalar, cuja equação de estado, p = wρ, varia dinamicamente,

iii

sendo possı́vel reproduzir, com maior liberdade, a história da composição do Universo. Existe uma

grande variedade de modelos de energia escura, sendo que a maioria difere na escolha do Lagrangiano

para o campo escalar e na respectiva interpretação fı́sica.

Rapidamente se tornou natural considerar que a componente de energia escura possa interagir com

a matéria convencional e com a matéria escura, dando origem a padrões observacionais. Usualmente, é

imposta uma função de acoplamento ao nı́vel das equações do movimento. No entanto, esta interacção

pode emergir naturalmente através do próprio campo escalar presente na teoria, que representa o papel

de energia escura. Isto é possı́vel através de uma transformação conforme/disforme do tensor da métrica,

levando a uma interacção descrita ao nı́vel do Lagrangiano.

As técnicas desenvolvidas no contexto da teoria de Sistemas Dinâmicos têm sido fulcrais para o de-

senvolvimento e interpretação de modelos cosmológicos. O seu uso permite não só descrever o nosso

Universo no passado e no presente (onde os resultados podem ser comparados com os dados observa-

cionais), mas também fazer previsões (ou especulações) acerca da sua evolução futura. Esta dissertação

baseia-se nessas mesmas técnicas para desenvolver modelos cosmológicos capazes de explicar a fase

de expansão acelerada do Universo que vivemos no presente, permitindo interacções entre a energia

escura e as restantes componentes de matéria/energia naturalmente presentes na teoria. Neste sentido,

na exploração dos modelos cosmológicos propostos, conjugam-se dois pontos de vista complementares:

faz-se uma análise dinâmica baseada em princı́pios matemáticos bem estabelecidos e uma análise das

consequências cosmológicas, baseadas em hipóteses motivadas e relevantes.

No Capı́tulo 1 começamos por fazer uma breve apresentação dos conceitos utilizados nos Capı́tulos

que se seguem. Primeiramente, é feita uma breve introdução à teoria matemática de Sistemas Dinâmicos

e às principais técnicas utilizadas ao longo deste trabalho. De seguida, apresenta-se uma pequena

introdução à teoria de Relatividade Geral e ao modelo padrão da Cosmologia. Finalmente, discute-se

o conceito de energia escura e as suas principais caracterı́sticas. É também feita a distinção entre o

campo escalar canónico, ou quintessência, e o campo escalar relativista, denominado taquião, no con-

texto de Teoria Quântica de Campo, com um paralelismo à teoria clássica. Terminamos com uma breve

revisão das diferentes aplicações cosmológicas destes dois campos, presentes na literatura.

No Capı́tulo 2 apresentamos o conceito de transformações conformes/disformes e é feita uma análise

do seu significado do ponto de vista matemático e fı́sico. De seguida mostramos como este tipo de

transformações pode ser usado para descrever modelos cosmológicos em diferentes referenciais com

interpretações fı́sicas especı́ficas. Neste sentido, somos naturalmente levados para a descrição onde

se permite interacções entre energia escura e as outras formas de matéria/energia presentes na teo-

ria. Discute-se as principais consequências cosmológicas dessa mesma interacção e apresenta-se uma

breve revisão dos diferentes tipos de acoplamentos considerados na literatura. Os acoplamentos prove-

iv

nientes de transformações conformes/disformes destacam-se por surgirem naturalmente numa teoria que

já contém um campo escalar, permitindo estudar um determinado modelo cosmológico num referencial

onde a interpretação fı́sica é mais evidente e/ou conveniente. A principal novidade associada aos mod-

elos acoplados é o aparecimento de pontos fixos, denominados scaling, que descrevem um Universo a

evoluir para um estado onde as densidades de energia escura e matéria escalam uma com a outra. Assim,

a existência de acoplamentos pode ter um papel benéfico, na medida em que estas soluções podem ser

usadas como forma de aliviar o problema da coincidência cosmológica, relacionado com a necessidade

de escolher condições iniciais especı́ficas para o nosso Universo de forma a obter a configuração que se

observa hoje.

No Capı́tulo 3 aplicamos as ideias descritas nos Capı́tulos anteriores a um modelo com um acopla-

mento conforme, onde o papel de energia escura é representado por um campo escalar taquiónico e se

admite um potencial quadrático inverso. Faz-se uma análise detalhada do ponto de vista dinâmico e

extraem-se as principais consequências cosmológicas. O estudo é feito em comparação com o modelo

desacoplado estudado anteriormente onde existe apenas um ponto fixo estável e capaz de descrever um

Universo em expansão acelerada, correspondente a um Universo a evoluir para um estado totalmente

dominado por energia escura. Por outro lado, o modelo acoplado, admite soluções de scaling, abrindo

portas para novas configurações. Com base na análise dinâmica e no estudo de estabilidade dos pontos

fixos encontrados, conclui-se que este modelo só é capaz de reproduzir a história da constituição do

Universo para condições iniciais muito particulares.

No Capı́tulo 4 implementamos um modelo com um acoplamento disforme, onde se toma um campo

escalar canónico para descrever a energia escura. Modelos de quintessência com um acoplamento

disforme já foram previamente considerados mas a análise existente na literatura é estendida, já que

se considera que a função disforme pode depender, não só do campo escalar, mas também das suas

derivadas temporais e/ou espaciais. Ainda que o sistema seja complexo e extensivo do ponto de vista

matemático, extraem-se as principais conclusões cosmológicas e, em particular, estuda-se em que medida

a dependência da função disforme no termo cinético do campo escalar afecta a dinâmica do sistema.

Terminamos com o Capı́tulo 5, onde referimos as principais conclusões do estudo realizado nesta

dissertação. Nomeadamente, discutimos as vantagens e desvantagens de considerar acoplamentos entre

energia escura e as outras formas de matéria. No futuro seria importante constranger os dois modelos

através dos dados observacionais disponı́veis. A análise detalhada da dinâmica de cada sistema permite

uma discussão das consequências cosmológica mais consistente, num tema tão pertinente e instigante

como a história da composição do Universo e a sua presente expansão acelerada.

Palavras-chave: Energia Escura, Sistemas Dinâmicos em Cosmologia, Transformações Con-formes/Disformes, Quintessência Acoplada, Taquião

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Abstract

Cosmology is the study of the universe, or cosmos, regarded as a whole. Standard cosmology began

with the formulation of the theory of General Relativity (GR) in 1915, by Albert Einstein. GR has

showed successful at describing the nature of our Universe, at both small and large scales. It has also

been capable of making rigorous predictions, which could only be confirmed later with the advent of

technology, as was the case with the recently-detected gravitational waves. For this reason, the standard

model of Cosmology is based on Einstein’s theory of gravitation and its cosmological implications.

However, in 1998, it was discovered that the Universe seems to be experiencing a period of accelerated

expansion, which can not be explained by any macroscopic type of matter, detected so far, included in

the Standard Model of Particle Physics. This breakthrough lead to the need of extending the present

theory of Gravitation.

Therefore, one of the main challenges in Cosmology concerns the determination of the composition

of the Universe. Taking only into account ordinary matter, such as radiation or non-relativistic matter,

which is gravitationally attractive (and never repulsive), there seems to be no reason to consider an

accelerated expanding scenario. Currently, one of the most accepted generalisations consists on assuming

that the acceleration is powered by an unknown source of energy/matter component, characterised by an

effective negative pressure. In reality, to fit the observational data, this source would have to be the

most abundant among the known constituents of the Universe. Such an unknown component is usually

classified under the broad heading of Dark Energy (DE). Additionally, the measurements of the rotation

curves of galaxies are not in agreement with the theoretical predictions based on Newtonian mechanics.

These observations suggest the presence of an undetected type of non-relativistic matter which seems to

neither emit nor absorb radiation and therefore can not be detected through electromagnetic interaction.

For this reason it is usually referred to as Cold Dark Matter (CDM).

Einstein himself, unknowingly, attempted to solve this problem, while aiming for a static cosmo-

logical solution, which called for a component with negative pressure. He chose to rule this rather odd

paradigm as non-physical and instead introduced his famous cosmological constant Λ to the gravitational

action. Even though this static solution was found to be unstable, we can rely upon Einstein’s thinking to

try to explain the accelerated expanding Universe. Thus, the simplest dark energy model is represented

by the cosmological constant Λ, which is now taken to be a cosmological source with pΛ = −ρΛ (where

p stands for the pressure and ρ for the energy density, which, for consistency reasons, is always assumed

to be positive). By adding a dark matter component we arrive at the current standard model of Cosmol-

ogy: the ΛCDM model. However, the ΛCDM model is known to present some conceptual issues. As an

attempt to avoid these problems, the cosmological constant is often generalised to a scalar field, whose

dynamical equation of state, p = wρ, could more naturally reproduce the evolution of the Universe.

vii

There is a wide variety of dark energy models which differ on the choice of the Lagrangian for the

scalar field and its corresponding physical interpretation. The interest in the possibility of having dark

energy interacting with the other matter/energy fluids present in the theory arose naturally. Customarily,

a coupling function is imposed at the level of the field equations. However, it could be more naturally

generated by means of the scalar field already present in the theory, through a conformal/disformal

transformation of the metric tensor, casting the interaction into a Lagrangian description.

The tools of Dynamical Systems have proved ideal for the development of a theoretical understanding

of the evolution of our Universe. Their use also allows for future predictions/speculations. In this thesis,

we focus on these tools in order to build viable cosmological models, while trying to explain the late-time

acceleration of the Universe through a coupled dark energy component. Hence, the work developed in

the context of this dissertation relies on two complementary approaches: a complete dynamical study

based on well-established mathematical principles and an analysis of the cosmological consequences

based on well-motivated and relevant physical hypotheses.

In Chapter 1 we begin with a brief overview of the main concepts included in the following Chapters.

We start with a succinct introduction to the theory of Dynamical Systems and the main techniques used

throughout this work. We follow to present some of the main aspects concerning the theory of General

Relativity and Cosmology. Finally, we introduce the concept of Dark Energy and discuss the conven-

tional scalar field descriptions in the context of Quantum Field Theory (while making a comparison with

the classical approach): the canonical scalar field, the so-called quintessence, and the relativistic scalar

field, the tachyon field. We also perform a brief review of the main applications of these scalar fields in

cosmology present in the literature.

In Chapter 2 we introduce the mathematical and physical formalisms regarding the concept of confor-

mal/disformal transformations. Next, we show how these ideas can be implemented in order to describe

cosmological models in different frames with distinct physical interpretations. This naturally leads to

the cosmological approach where dark energy is allowed to interact with other matter/energy sources

present in the theory. We discuss the most important cosmological consequences of such an interaction

and perform a brief review on the different couplings considered in the literature. Couplings emerging

from conformal/disformal transformations can be accomplished through a fundamental scalar field al-

ready present in theory. They are of paramount importance for cosmological models by allowing for

the study to be made in a specific frame where the physical interpretation is more evident/convenient.

The main novelty associated with these models lies in the emergence of new fixed points, referred to

as scaling fixed points. These solutions describe a Universe evolving towards a state where the energy

densities of dark energy and coupled matter scale with each other. Hence, the presence of the coupling

could yield favourable results, for instance it could alleviate the cosmic coincidence problem, related to

viii

the need of postulating specific initial conditions for the Universe in order to reproduce the configuration

which we observe today.

In Chapter 3 we discuss the application of the ideas presented in the previous Chapters to a con-

formally coupled model where the role of dark energy is played by a tachyon field, characterised by

an inverse square potential, which is allowed to interact with the matter sector. A detailed dynamical

analysis of the cosmological outcome is performed in comparison with the previously studied uncoupled

tachyonic dark energy model. In the latter, there exists only one stable critical point capable of describ-

ing the late time acceleration of the Universe, corresponding to a totally dark energy dominated future

configuration. The conformally coupled model, on the other hand, provides scaling solutions, allowing

for different frameworks. Based on the dynamical analysis, we conclude that this model is only capable

of reproducing the history of the Universe for a specific set of initial conditions.

In Chapter 4 we implement a disformally coupled model where dark energy is represented by a

canonical scalar field with an exponential potential. The analysis in the existing literature is extended by

assuming that the disformal coefficient depends both on the scalar field and its kinetic term (related to

time and/or spatial derivatives of the field). Even though this is a complicated and mathematically exten-

sive model, we extract its main cosmological features and, in particular, we study how the dependence

of the transformation on the kinetic term affects the dynamics of the system.

We conclude in Chapter 5 with some final remarks regarding the work presented in this thesis.

Namely, we discuss the advantages/disadvantages of considering couplings between dark energy and

the matter sector. In the future it would be important to use observational data to constrain the models,

at the level of the background and by means of perturbation theory. The detailed dynamical analysis

performed for each system provides a better understanding of the cosmological consequences and the

physically allowed configurations, improving the consistency level of the study.

Keywords: Dark Energy, Dynamical Systems in Cosmology, Conformal/Disformal Transforma-tions, Coupled Quintessence, Tachyon

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Introduction 1

1.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Linear Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Basics of General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Standard Model of Cosmology - The FLRW model . . . . . . . . . . . . . . . . . . . . 12

1.4 Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Quintessence Field and Tachyon Field . . . . . . . . . . . . . . . . . . . . . . . 18

2 Conformal and Disformal Transformations 21

2.1 Lagrangian Formalism of GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Conformal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Disformal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Einstein Frame and Jordan Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Interacting Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Conformally Coupled Tachyonic Dark Energy 35

3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Background Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Phase Space and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Dynamical System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xi

3.5.1 Fixed Points, Stability and Phenomenology . . . . . . . . . . . . . . . . . . . . 44

3.5.2 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.3 Physical Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Viable Cosmologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 Effective Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Disformally Coupled Quintessence 59

4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Background Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Phase Space and Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Dynamical System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5.1 Fixed points, Stability and Phenomenology for a pressureless fluid . . . . . . . . 68

4.5.2 Fixed points, Stability and Phenomenology for a relativistic fluid . . . . . . . . . 73

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Final Remarks 79

Bibliography 81

A Natural Units 97

B Conformally Coupled Tachyonic Dark Energy - Linear Stability Matrix 100

C Conformally Coupled Tachyonic Dark Energy - Eigenvalues of the Stability Matrix 101

D Disformally Coupled Quintessence - Equation of Motion 103

E Disformally Coupled Quintessence - Interaction term 105

F Fixed Points for the Disformally Coupled System with no Kinetic Dependence 107

G Disformally Coupled Quintessence - Eigenvalues of the Stability Matrix 109

xii

List of Tables

3.1 Couplings of a barotropic perfect fluid to a tachyonic field studied in the literature . . . . 42

3.2 Fixed points and corresponding existence conditions and cosmological parameters for

the conformally coupled tachyonic model . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Effective equation of state parameter for the conformally coupled tachyonic model . . . 46

3.4 Dynamical stability for the fixed points of the conformally coupled tachyonic model . . . 48

4.1 Fixed points and corresponding cosmological parameters for disformal quintessence cou-

pled to a non-relativistic fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Effective equation of state parameter for disformal quintessence coupled to a non-relativistic

fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Fixed points for disformal quintessence coupled to a relativistic fluid with a linear de-

pendence of the disformal function on the kinetic term . . . . . . . . . . . . . . . . . . 74

4.4 Effective equation of state parameter for disformal quintessence coupled to a relativistic

fluid with a linear dependence of the disformal function on the kinetic term . . . . . . . 74

F.1 Fixed points for disformal quintessence coupled to a fluid with an arbitrary constant

equation of state for the case where the disformal function does not depend on the kinetic

term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

G.1 Eingenvalues of the fixed points for disformal quintessence coupled to a non-relativistic

fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

G.2 Eingenvalues of the fixed points for disformal quintessence coupled to a relativistic fluid 110

xiii

List of Figures

1.1 Saddle node and transcritical bifurcation diagrams . . . . . . . . . . . . . . . . . . . . . 8

3.1 Bifurcation diagram and attractor of the coupled tachyonic system according to the pa-

rameter region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Phase portrait projections for the uncoupled and coupled tachyonic models . . . . . . . . 52

3.3 Viable cosmologies with fine tuning for the conformally coupled tachyonic model . . . . 53

3.4 Example of the evolution of the relative energy densities and the EoS parameters accord-

ing to the conformally coupled tachyonic model . . . . . . . . . . . . . . . . . . . . . . 54

3.5 Effective potential for the conformally coupled tachyonic model . . . . . . . . . . . . . 56

4.1 Parameter region for a stable fixed point for the case of quintessence disformally coupled

to a non-relativistic fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Attractor of the disformally coupled system according to the parameter region . . . . . . 73

xv

Chapter 1

Introduction

In this Chapter we give a brief introduction to some important topics and mathematical techniques

which will be relevant throughout this work. We start with a succinct introduction to the mathematical

theory of Dynamical Systems, then we discuss some important concepts regarding the theory of General

Relativity and its implications to Cosmology and finally, introduce the concept of Dark Energy and how

one implements this ideas in order to construct a cosmological model.

1.1 Dynamical Systems

This section is mainly based on references [1–5].

A dynamical system is one whose state varies with time, t. Whenever t is taken to be continuous the

dynamics is customarily described by a set of differential equations,

dx1dt

= ẋ1 = f1(t, x1, ..., xn),

... (1.1)

dxndt

= ẋn = fn(t, x1, ..., xn),

where t ∈ R, n is the dimension of the state space X ⊆ Rn and x = (x1, ..., xn) ∈ X is an element of

the state space which represents the state of the system. The function F = (f1(x), ..., fn(x)) is a vector

field that characterises the specific system which is being studied, with F : X → Rn.

Ordinary differential equations of the form ẋ = F(x), which do not depend explicitly on time,

are called autonomous or time independent as opposed to ordinary differential equations which depend

explicitly on time, ẋ = F(x, t), and are referred to as non-autonomous or time dependent.

The state space can be defined as the set of all possible values of the quantities that must be used

to fully describe the state of the system through time. One can think of ecological models, e.g. Lotka-

Volterra models, where the state space corresponds to the possible number of elements in each pop-

1

ulation. When working with cosmological models suitable normalised quantities are often chosen to

describe the state of the Universe and this will be studied in greater detail further on.

We will then have a set of n first order, ordinary differential equations describing how the system

evolves in time. A solution (also referred to as trajectory or phase curve) of the dynamical system (1.1)

on Rn is any function ψ : I ⊆ R → Rn which satisfies ψ̇(t) = F(ψ(t)) for a certain time interval

I . The image of a solution ψ(t) in Rn is often called an orbit or a trajectory in the phase space. The

corresponding physical system will evolve in time according to the motion of x ∈ Rn along that orbit of

the dynamical system.

Also, a dynamical system of the form (1.1) is said to be linear if all the x1, ..., xn appear linearly, i.e.,

to the first power only, on the RHS of the system of equations.

Theorem 1.1.1 (Existence and Uniqueness). Consider a general autonomous equation ẋ = F(x), x ∈

Rn, with initial value x0 ∈ Rn. If F : Rn → Rn is continuously differentiable, then for every x0 ∈ Rn,

there exists a maximal time interval I and a unique function ψ, defined on I , such that

ψ̇(t) = F(ψ(t)), ψ(0) = x0.

(Proof in [3], pages 161-167).

It can be shown that, in some specific cases, the maximal interval I for which the Existence and

Uniqueness Theorem is valid, can be extend for all t ∈ R. (For proof see [3], pages 171-173).

Theorem 1.1.1 tells us that, given two solutions of the dynamical system ψ(t) and ϕ(t) with ψ(0) =

ϕ(t0), then, by uniqueness of the solutions, ψ(t) = ϕ(t + t0). This means that we can distinguish a

solution by a particular point in the state space that it passes through at a specific time. Without loss

of generality, taking t0 = 0 and x0, we denote by x(t; x0), the solution for which ψ(0) = x0. This is

referred to as giving an initial condition or initial value and so, in fact, Theorem 1.1.1 states that solutions

can be labelled by their initial conditions.

Definition 1.1.1 (Orbit). Consider a general autonomous equation ẋ = F(x), x ∈ Rn. Let x0 ∈ Rn be a

point in the phase space. The orbit through x0,O(x0), is defined as the set of points in the state space that

lie on the trajectory which passes through x0. More precisely: O(x0) = {x ∈ Rn : x = x(t; x0), t ∈ I},

where I ⊆ R is the maximal time interval.

This means that an orbit O(x0) is the graph of a solution of the differential equation starting from the

initial condition x0. The collection of all the qualitatively different trajectories of the system represents

the phase portrait.

Definition 1.1.2 (Fixed point). Consider a general autonomous equation ẋ = F(x), x ∈ Rn. This

equation is said to have a fixed point at x = x∗ if and only if F(x∗) = 0.

Fixed points are also referred to as equilibrium points, stationary points or critical points.

2

Definition 1.1.2 implies that x∗ corresponds to a solution that does not change in time, which is to

say that the velocity on the phase space is zero, F(x∗) = 0. The question of whether the system, when

perturbed, will remain close to this state leads to the definition of stability, i.e., of the local properties of

the orbits in the neighbourhood of the fixed point.

Definition 1.1.3 (Lyapunov stability of a fixed point). Let x∗ be a fixed point of the system ẋ = F(x),

x ∈ Rn. x∗ is said to be stable (or Lyapunov stable) if, given some � > 0, there exists a δ > 0 such that,

for any other solution of the system ψ(t), satisfying | x∗ − ψ(t0) |< δ, then | x∗ − ψ(t) |< � for t > t0.

A solution which is not stable will, of course, be unstable and at least some solutions starting nearby

will move away from it. Whenever the fixed point is found to be stable and every solution approaches

the fixed point for any nearby initial conditions then it is called asymptotically stable.

Definition 1.1.4 (Asymptotic stability of a fixed point). Let x∗ be a fixed point of the system ẋ = F(x),

x ∈ Rn. x∗ is said to be asymptotically stable if it is Lyapunov stable and, for any other solution of the

system ψ(t), there exists a constant δ such that, if | x∗ − ψ(t0) |< δ, then limt→∞ | x∗ − ψ(t) |= 0.

The main difference between Definition 1.1.3 and Definition 1.1.4 is that, near an asymptotically

stable fixed point, as t → ∞, all trajectories will eventually approach it. On the other hand, for a

Lyapunov stable fixed point, we only know that solutions starting sufficiently close to the fixed point will

remain close for all time but there is no guarantee that they will approach the fixed point as they could,

for instance, simply orbit around the fixed point.

Both these definitions apply only to autonomous systems since in non-autonomous systems δ and �

could, and in principle would, have explicit dependence on time and more care should be taken. However

throughout this work we will only focus on autonomous dynamical systems. Also, since most fixed points

in cosmological models are asymptotically stable, we will make no distinction between Lyapunov stable

and asymptotically stable, unless it is specifically needed.

Definition 1.1.5 (Heteroclinic orbits). Let x∗ be a fixed point of the system ẋ = F(x), x ∈ Rn.

A heteroclinic orbit is an solution ψ(t) for which there exist two fixed points x− and x+ such that

limt→−∞ ψ(t) = x− and limt→+∞ ψ(t) = x+. This simply means that an heteroclinic orbit is an orbit

connecting distinct fixed points. If the orbit connects one fixed point to itself it is called a homoclinic

orbit.

The concept of an invariant set plays a crucial role in the theory of dynamical systems.

Definition 1.1.6 (Invariant set). Consider a general autonomous equation ẋ = F(x), x ∈ Rn and let

S ⊂ Rn be a set. In this context, S is said to be invariant if for all x0 ∈ S we have x(t; x0) ∈ S for all

t ∈ R. If t ≥ 0 (t ≤ 0) then S is said to be a positively (negatively) invariant set. In other words, all

trajectories starting in the invariant set, will never leave the invariant set.

3

Definition 1.1.6 also asserts that an invariant set is some part of the state space which is not connected

to the rest of the state space by any orbit. Also, we can assert that if S is an invariant set and x0 ∈ S then

the orbit O(x0) belongs to S. This means that an invariant set can also be defined as a union of orbits.

Until now we have defined the mathematical concept of different types of stability but to study the

stability properties of a specific fixed point we need a methodology. For this purpose we introduce the

so-called linear stability theory which allows for a good physical interpretation of most cosmological

models.

1.1.1 Linear Stability Theory

We consider a general autonomous equation ẋ = F(x), x ∈ Rn. In order to determine the stability

of a given fixed point x∗(t) we need to understand the nature of trajectories close to the fixed point.

For this purpose, we linearise the system around the fixed point, which is to say that, assuming F(x) =

f1(x), ..., fn(x) to be continuously differentiable, we Taylor expand each fi(x) around the fixed point:

fi(x) = fi(x∗) +n∑j=1

∂fi∂xj

(x∗)uj +1

2!

n∑j,k=1

∂2fi∂xj∂xk

(x∗)ujuk + ..., (1.2)

where u = u1, ..., un is defined as u = x − x∗ and, of course, fi(x∗) = 0. The first order truncation of

equation (1.2) is called the linearisation of the differential equation at the fixed point x∗ and is a good

first approximation to the full system near x = x∗:

fi(x) ≈n∑j=1

∂fi∂xj

(x∗)uj . (1.3)

So, it is reasonable to expect that the behaviour of the linearisation at x = x∗ is a good approximation

of the behaviour of the non-linear system near x = x∗.

Therefore, an important object for linear stability theory is the stability matrix,M:

M =[∂fi∂xj

]=

∂f1∂x1

· · · ∂f1∂xn...

. . ....

∂fn∂x1

· · · ∂fn∂xn

. (1.4)

The stability matrix is an n × n matrix (where n is the dimension of the phase space) containing

information about each first order derivative of each function fi:Mij = ∂fi∂xj1. Accordingly, there are n

eigenvalues of this matrix which, when evaluated at the fixed point, will provide the information about

the stability of the system.

1The stability matrix is the Jacobian matrix of traditional vector calculus.

4

Theorem 1.1.2. Let x∗ be a fixed point of the system ẋ = F(x), x ∈ Rn and suppose that all of the

eigenvalues of the stability matrix evaluated at the fixed point,M(x∗), have negative real parts. Then,

the fixed point x∗ is asymptotically stable.

The proof for Theorem 1.1.2 can be found in [2] (pages 24, 25).

This gives us a tool to determine the stability of a specific fixed point. As a consequence we have

three distinct cases:

• If all of the eigenvalues ofM(x∗) have negative real parts then the fixed point is asymptotically

stable and is said to be an attractor.

• If all of the eigenvalues ofM(x∗) have positive (non-zero) real parts then the fixed point is said to

be a repeller.

• Finally, if at least two of the eigenvalues ofM(x∗) have non-zero real parts with opposite sign then

the fixed point is said to be a saddle point, meaning that it attracts trajectories in some directions

but repels them along others.

The previous characterisation only gives information about the stability of a fixed point given that

none of the eigenvalues ofM(x∗) have real part equal to zero. This leads us to the definition of hyper-

bolic fixed point:

Definition 1.1.7 (Hyperbolic fixed point). Let x∗ be a fixed point of the system ẋ = F(x), x ∈ Rn and

consider the stability matrix evaluated at the fixed point, M(x∗). Then x∗ is called a hyperbolic fixed

point if none of the eigenvalues ofM(x∗) have zero real part 2.

Stability properties can only be derived from linear stability theory whenever the fixed points in

study are hyperbolic, meaning that it fails for non-hyperbolic points and their stability properties should

be studied with alternative methods [2].

1.1.2 Bifurcations

Until now, we have only considered dynamical equations of the form (1.1) depending on a set of

dynamical variables x = x1, ..., xn. But besides dynamical variables, the dynamical equations of a

model can contain time-independent quantities. The value of these parameters is fixed for a specific

application of the model. For example, for an ecological model, one useful parameter would be the

growth rate of a specific population. When we wish to leave the parameter free, it is advantageous to

think of it as a continuous variable which is time-independent. The result is a set of dynamical equations

indexed by that parameter.2Note that this definition is made according to the linearised system, as hyperbolicity is a much broader concept [3].

5

Following this, consider the parametrised vector field:

ẋ = F(x;λ), x ∈ Rn, λ ∈ Rp, (1.5)

where F : Rn → Rn is a continuous differentiable function and again, n is the dimension of the system,

and p is the number of parameters: λ = λ1, ..., λp. Now suppose that the family (1.5) has a fixed point at

(x;λ) = (x∗;λ∗), i.e., F(x∗;λ∗) = 0. We wish not only to ask if this fixed point is stable or not but we

also want to know how its stability (or instability) will be affected as λ is varied.

If the fixed point is hyperbolic, according to Definition 1.1.7, we know that its stability can be deter-

mined by the sign of the eigenvalues of the stability matrix in the linear approximation. Indeed, when

F(x∗;λ∗) = 0 andM(x∗;λ∗) has no eigenvalues with zero real-part,M(x∗;λ∗) is an invertible matrix

and, by the Implicit Function Theorem, for λ sufficiently close to λ∗, there exists a unique function,

x(λ), such that F(x(λ);λ) = 0. By continuity of the eigenvalues with respect to the parameters, for λ

sufficiently close to λ∗,M(x(λ);λ) has no eigenvalues with zero real-part. Therefore, for λ sufficiently

close to λ∗, the hyperbolic fixed point (x∗;λ∗) of (1.5) persists and its stability type remains unchanged.

To summarise, in a neighbourhood of λ∗, an isolated fixed point of (1.5) persists and always has the same

stability type.

Therefore, the question of whether the stability and number of fixed points is changed when λ varies

is only relevant for non-hyperbolic fixed points. In this case, for λ near λ∗ (and for x close to x∗), the

dynamical behaviour could be completely altered.

Definition 1.1.8 (Bifurcation of a Fixed Point). A fixed point (x;λ) = (x∗;λ∗) of a parameter family of

n-dimensional vector fields is said to undergo a bifurcation at λ = λ∗ if the flow for λ near λ∗ and x near

x∗ is not qualitatively the same as the flow near x = x∗ at λ = λ∗. In this case, x∗ is called a bifurcation

point and the parameter value λ = λ∗ a bifurcation value.

Note that the condition that a fixed point is non-hyperbolic is a necessary but not sufficient condi-

tion for bifurcation to occur in parameter families of vector fields. Bifurcations are usually classified

according to how the stability and number of fixed points are changed.

As an example, in the simplest case, the one dimensional system,

dx

dt= f(x;λ), t > 0 (1.6)

where λ is a real parameter and f is some continuously differentiable function of x and λ. The fixed

points x∗ of equation (1.6) are found by setting f(x;λ) = 0. If fx ≡ ∂f∂x = 0 at λ = λ∗, then several

fixed points may exist corresponding to one single value of λ, in a neighbourhood of λ∗. This happens

because the Implicit Function Theorem does not apply when ∂f∂x (x∗;λ∗) = 0.

6

The representation of f(x;λ) = 0 is called the branching diagram. The intersecting branches are

named bifurcating solutions and the points of intersection (in which stability changes), are called bifur-

cation points.

In our definition, the bifurcation value λ∗ is defined according to,

fx(x∗;λ∗) = 0, f(x∗;λ∗) = 0.

But, by the Implicit Function Theorem, f(x∗;λ) = 0 implies λ = λ(x∗) whenever fx(x∗;λ) 6= 0, This

can be expressed by differentiating f according to x∗,

fx∗ + fλdλ

dx∗= 0. (1.7)

Equation (1.7) shows that, if fλ 6= 0, then at a bifurcation point where fx∗ = 0, dλdx∗ = 0.

Example 1.1.1 (Saddle-node bifurcation). Consider the one-dimensional quadratic equation,

dy

dt= f(y;λ) = λ− y2, (1.8)

for λ ≥ 0. Solving equation (1.8) equal to zero translates into f(y;λ) = 0, which renders the fixed

points,

y∗1 =√λ, y∗2 = −

√λ (1.9)

The bifurcation point λ = 0 gives the intersection of the two branches of fixed points whose existence

is allowed near λ = 0 and y = 0 by the Implicit Function Theorem as dfdy = −2y vanishes at y = 0,

where the fixed points collide giving rise to a single non-hyperbolic fixed point. Besides, dfdy (y∗1 ;λ) =

−2y∗1 = −2√λ and dfdy (y∗2 ;λ) = −2y∗2 = 2

√λ.

This defines the stability of each fixed point. For λ > 0, y∗1 (represented by the solid branch of the

parabola in Figure 1.1 (a)) is stable and y∗2 (represented by the dashed branch of the parabola in Figure

1.1 (a)) is unstable. As y∗1 and y∗2 always have opposite signs they will also have opposite stability

characters. Hence, one single branch of fixed points experiences a transition from stable to unstable and,

in particular, there is an exchange of stability at the bifurcation point y∗ = 0. Note that, at the bifurcation

point, dfdλ = 1 6= 0 and equation (1.7) impliesdλdy∗

= 0 there. As λ is varied, the two critical points y∗1

and y∗2 get closer together, collide and are mutually destroyed.

This is a standard example of a saddle-node bifurcation which represents the procedure by which

fixed points are “created” or “destroyed”.

Example 1.1.2 (Transcritical bifurcation). Consider the one-dimensional logistic equation,

7

λ

y*

(a) Saddle node bifurcation

λ

n*

(b) Transcritical bifurcation

Figure 1.1: Bifurcation diagrams for the saddle node bifurcation described in Example 1.1.1 (left) andthe transcritical bifurcation described in Example 1.1.2 (right). A solid curve is used to represent a familyof stable fixed points whereas a dashed curve is used to highlight a family of unstable fixed points. Thevertical lines represent the flow generated by each system along the vertical direction.

dn

dt= f(n;λ) = λn− n2, (1.10)

where n represents the number of individuals in a given population. The parameter λ describes the

evolution of the population. If λ is relatively small then the population grows (dies) exponentially if

λ > 0 (λ < 0). If λ is big, the population grows too fast and eventually, it will become so large that

the growth rate drops due to lack of food income. The second term in (1.10), being non-linear, gives a

natural saturation of the exponential population growth.

The fixed points on equation (1.10) are found through f(n;λ) = 0:

n∗1 = 0, n∗2 = λ. (1.11)

These two fixed points coincide at λ = 0. This means that these two branches of fixed points

intersect at the bifurcation point λ = 0. Near λ = 0 and n = 0 the existence of these two branches is

allowed by the Implicit Function Theorem, because dfdn = 0 is when n = 0 and λ = 0, highlighting the

non-hyperbolic character of this fixed point.

Actually, we have,

df

dn(n∗1 , λ) = λ,

df

dn(n∗2 , λ) = −λ. (1.12)

By linear stability theory, the fixed point n∗1 is stable ifdfdn(n∗1 , λ) < 0 and the fixed point n∗2 is

stable if dfdn(n∗2 , λ) < 0. This translates into n∗1 being stable for λ < 0 and unstable for λ > 0 and

8

n∗2 being stable for λ > 0 and unstable for λ < 0. Hence, whenever n∗1 is stable n∗2 is unstable and

vice-versa. The two branches have exact opposite stabilities and exchange stability at the bifurcation

point λ = 0. A depiction of this behaviour can be found in Figure 1.1 (b).

Also, here, fλ = n = 0 at the bifurcation point, suggesting that, by equation (1.7), dλdn∗ could be

different from zero at this point. This is a typical example of a transcritical bifurcation and the logistic

equation is the canonical form of transcritical bifurcation.

1.2 Basics of General Relativity

General Relativity (GR), formulated in 1915 [6], is Albert Einstein’s theory of space, time and grav-

itation. It stands for a mathematical description of the three spatial dimensions plus one time dimension,

through four dimensional manifolds. This means that the three spatial dimensions and time are features

of a fundamental four-dimensional spacetime. This was soon considered to be a powerfully predictive

theory when it passed rigorous observational tests [7], namely, the recently detected gravitational waves

[8]. In GR, gravity is a manifestation of the curvature of spacetime itself, making it a purely geometrical

theory: locally, mass distorts spacetime, giving rise to a gravitational field, and on cosmological scales,

spacetime itself can be curved.

General Relativity rises as a generalisation of Einstein’s Special Relativity (SR) [9] in order to obtain

a coherent theory of gravitation. In GR the notions of distance and time between two spacetime points

in the manifold are encoded in the metric tensor, gµν , which is a function of the spacetime coordinates

xµ. In other words, the metric fixes the causal structure of spacetime (the light cones). For the four

dimensional manifold we take µ = 0, 1, 2, 3 and latin indices, i = 1, 2, 3, are used to denote spatial

coordinates.

We can define the determinant of the metric, g ≡ det(gµν) and the inverse of the metric, gµν , which,

given g 6= 0 (i.e., gµν invertible), is fully determined by the condition: gµαgαν = δνµ, where the symmetry

of gµν implies symmetry on the inverse, gµν . This means that the metric tensor and its inverse can be

used as a tool to raise or lower indices of a tensor, for example:

Xµ = gµνXν . (1.13)

Throughout this work we will rely on the Einstein notation, which states that when some index

appears twice in a single term, a summation of that term over all the values of the index is implied. For

example, for the case of the term in (1.13):

Xµ = gµνXν =3∑

ν=0

gµνXν . (1.14)

9

The separation between two points is defined according to the line element, ds,

ds2 = gµνdxµdxν , (1.15)

and is given by

`AB =

∫ BAds, (1.16)

where dxµ is an infinitesimal displacement vector in the direction of xµ.

In a curved spacetime, the generalisation of the partial derivative is the covariant derivative:

∇µXα = ∂µXα − ΓβµαXβ, ∇µXα = ∂µXα + ΓαµβXβ, (1.17)

respectively for covariant and contravariant tensors. In the previous expressions, Γλµν are the connection

coefficients, related to the parallel transport (transporting a given vector along a curve γ without any

change to its length or direction so as to obtain a parallel vector at each point of γ).

In order for the scalar product to stay invariant through parallel transport along any curve (which is to

say that its covariant derivative must vanish), we ask for compatibility between the covariant derivative

and the metric,

∇αgµν = 0. (1.18)

Assuming that the spacetime connection lower indices are symmetric and, according to the definition

of covariant derivative given in equation (1.17), this implies

Γλµν =1

2gλδ(∂νgδµ + ∂µgδν − ∂δgµν), (1.19)

which is the Levi-Civita connection, also referred to as Christoffel symbols of the second kind.

In the theory of General Relativity there is a very clear connection between the geometry of the

manifold, encoded in the metric tensor gµν , and the entire matter content of the Universe, expressed

by the so-called energy-momentum tensor, Tµν (EM). This relation is described by the Einstein field

equations (without a cosmological constant):

Gµν = Rµν −1

2gµνR =

8πG

c4Tµν , (1.20)

where G is the classical Newton’s gravitational constant, c is the speed of light in vacuum, Gµν is the

Einstein tensor, which encapsulates all curvature information, R and Rµν are the Ricci scalar and Ricci

tensor, defined in terms of the Riemann tensor Rρµσν , as follows:

Rρµσν = ∂σΓρνµ − ∂νΓρσµ + Γ

ρσλΓ

λνµ − Γ

ρνλΓ

λσµ, (1.21)

10

Rµν = Rλµλν , (1.22)

R = gµνRµν . (1.23)

Given this, expression (1.20) corresponds to a set of 16 equations. Actually, by assuming that the

Einstein tensor and the energy momentum tensor are symmetric we are left with a set of 10 equations.

Additionally, it can be shown that the Riemann curvature tensor also satisfies a set of differential identities

called the Bianchi identities,

∇λRρµσν +∇νRρµλσ +∇σR

ρµνλ = 0, (1.24)

which reduces the Einstein equations to a set of 6 independent non-linear differential equations. From

(1.24) it is possible to extract four conservation laws, commonly termed the contracted Bianchi identities:

∇µGµν = 0 =⇒ ∇µTµν = 0 , (1.25)

which can be interpreted as generalisations of the classical energy and momentum conservation laws.

The Riemann tensor describes the spacetime curvature and is given in terms of the connections,

which define the spacetime geodesics (trajectories of free particles). In GR, a free particle’s trajectory is

the generalisation of a straight line in a curved spacetime and is given by the geodesic equation

d2xα

ds2+ Γαµν

dxµ

ds

dxν

ds= 0, (1.26)

where ds is defined in equation (1.15). Equation (1.26) with Γαµν given by (1.19) is the geodesic equation

for Euclidean space. There are some privileged parameters u for which the geodesic equation has the

form d2xα

du2+ Γαµν

dxµ

dudxν

du = 0. These are known as affine parameters. For an affine parameter, ds/du is

constant, so one is taken along the geodesic at a constant sort of rate. ds2 contains information about the

causal structure of the spacetime, in the sense that any non-zero vector V µ is described as:timelike

null

spacelike

if gµνV µV ν

< 0

= 0

> 0.

(1.27)

For timelike structure, the length is called proper time whereas for spacelike structure it is termed proper

distance. It can be shown that a massive particle follows a timelike path through spacetime (and in

particular a free particle follows a timelike geodesic) whereas massless particles, such as photons, follow

a null geodesic (the tangent vectors to its path are null). We can not construct spacelike paths between

11

two events since, as they are spatially separated, the proper time is not defined (there is no worldline

connecting the events).

Note that the proper time for an observer is given by dτ2 = ds2/c2, implying that, sometimes, this

is a useful affine parameter to parametrise the geodesics. In General Relativity, Newton’s first law of

motion can be translated as “free particles follows geodesics in spacetime”. In fact, if we think of a local

inertial coordinate system, where we may neglect the terms proportional to Γαµν , the geodesic equation

(1.26) for the affine parameter τ reduces to d2xα/dτ2 = 0. For non-relativistic speeds dτ/dt ' 1 and

the geodesic equation yields d2xi/dt2 = 0 (i = 1, 2, 3), which is identified as the Newtonian equation

of motion of a free particle.

Given a specific metric, both the structure of spacetime and the motion of particles within it can be

deduced.

For more details regarding the formal geometrical details of General Relativity we refer the reader

to, for example, [9–14].

1.3 Standard Model of Cosmology - The FLRW model

This section is mainly based on [15–19].

Cosmology is the study of the universe, or cosmos, regarded as a whole. Modern theoretical cosmol-

ogy relies mainly on Einstein’s theory of gravitation and its cosmological implications [20].

There are no general solutions to the Einstein field equations, (1.20), but, among others, Einstein

realised that, in order for the field equations to give a coherent description of the Universe, some as-

sumptions were needed. The first simplification is that the Universe should look the same at each point

and in all directions. This is the Copernican Principle, commonly generalised to the Cosmological Prin-

ciple, which states that space can be assumed to be spatially homogeneous and isotropic on sufficiently

large scales. That is to say that there are no special positions in the Universe and, so, the cosmological

metric, needs to be one which describes a time varying universe and also one that is, at each time, spa-

tially homogeneous and isotropic. Taking only geometrical arguments, the most generic spacetime for a

homogeneous and isotropic Universe with matter uniformly distributed, as a perfect fluid is given by the

Friedmann-Lemaı̂tre-Robertson-Walker metric [21–23] and has the following form:

ds2 = −dt2 + a2(t)(

dr2

1− kr2+ r2dθ2 + r2sin2θ , dϕ2

), (1.28)

where t is the cosmic time, k ∈ {−1, 0,+1} is the spatial curvature and a(t) > 0 is the scale factor

which describes the expansion or contraction of the universe (defined in a way such that a(ttoday) = 1).

Hereafter we will choose units such that c = 1 (see Appendix A for more details regarding the system of

units). For k = 1 the universe is said to be spatially closed, spatially open for k = −1 and, if k = 0, it

12

is spatially flat. The set of coordinates (r, θ, ϕ) are called the comoving coordinates, i.e., the coordinates

for which r, θ, ϕ are constant throughout time evolution. For cosmological applications it will be useful

to consider the homogeneous, isotropic, spatially flat3 (k = 0) FLRW metric (1.28), which in Cartesian

coordinates reads

ds2 = −dt2 + a2(t)δijdxidxj , (1.29)

Secondly, as a consequence of the cosmological principle, the various matter-energy components of

the Universe are assumed to be well described, at large scales and with high precision, by a continuous

perfect fluid, to which spacial comoving coordinates are assigned. In particular, the corresponding EM

tensor is fully described by its energy density ρ(t) and isotropic (no shear nor viscosity) pressure p(t).

By assuming a perfect fluid, the energy-momentum tensor can be written as:

Tµν = (p+ ρ)uµuν − pgµν , (1.30)

where uµ is the fluid’s four-velocity defined as

uµ =dxµ√−ds2

, (1.31)

which satisfies uµuµ = −1 and, for a comoving observer, is given by uµ = (−1, 0, 0, 0). Given this, and

taking the metric in (1.29), Tµν is a purely diagonal tensor:

Tµν =

−ρ 0 0 00 p 0 0

0 0 p 0

0 0 0 p

(1.32)

When p and ρ can be related through an equation of state (EoS), p ≡ p(ρ), we speak of barotropic

fluids. This is usually considered to be a linear relation: p = wρ, where w is termed the equation of

state parameter. It is well-known that for a non-relativistic (dust-like) perfect fluid w = 0, while for a

relativistic (radiation-like) fluid w = 1/3. The Standard Model of Particle Physics excludes fluids with

w /∈ [0, 1] though some phenomenological models need to rely on non-physical values of w in order to

explain the astronomical observations.

The dynamical equations arising from the Einstein field equations (1.20) assuming a FLRW metric

(1.29) and possible spatial curvature k, consist of two coupled differential equations for the scale factor

a(t) and the functions ρ(t) and p(t). The Friedmann constraint follows from the time-time component

of the Einstein field equations and can be written as:3The hypothesis that the Universe is flat has been argued in the context, for instance, of CMB results, presenting a sharp

feature in the temperature anisotropy spectrum on the very angular scale predicted for a spatially flat Universe [24].

13

H2 =

(ȧ

a

)2=

8πG

3ρ− k

a2, (1.33)

where H ≡ ȧa is the Hubble parameter (with the convention that an over-dot denotes differentiation with

respect to t).

On the other hand, from the spatial (diagonal) components of the Einstein field equations we can

derive:

ä

a= −4πG

3(ρ+ 3p), (1.34)

which is known as the Raychaudhuri equation. This last equation is of great value as it gives a straight-

forward condition on the parameters in order to draw a distinction between a universe with an increasing

expansion rate, if ä > 0, or decreasing, if ä < 0. These two cases usually correspond to a Universe

undergoing accelerated or decelerated expansion respectively. According to equation (1.34), it is clear

that if ρ+ 3p > 0 the Universe must be decelerating whereas if ρ+ 3p < 0 the Universe is accelerating.

The positive inequality is shown to correspond to the strong energy condition [13]. By assuming a linear

equation of state all of this information can be translated as conditions imposed on the EoS parameter:

w > −1/3 for deceleration and w < −1/3 for acceleration. This gives constraints regarding the phys-

ically relevant solutions and one can note that ordinary matter (the one we are used to experience as

baryons and radiation), which presents 0 ≤ w ≤ 1/3 cannot be used to power an accelerating Universe.

From the conservation of the EM tensor, (1.25), and from equations (1.30), (1.33) and (1.34), we can

derive the continuity equation:

ρ̇+ 3H(ρ+ p) = 0, (1.35)

which expresses energy conservation throughout the evolution of the Universe. Taking into account the

EoS parameter and solving for ρ:

ρ ∝ a−3(w+1). (1.36)

Considering that for matter, i.e., a dust-like fluid (baryonic matter and dark matter4), pm = 0⇒ wm = 0,

and for a radiation-like or ultrarelativistic fluid (photons and neutrinos), pr = ρr/3 ⇒ wr = 1/3, we

have ρm ∝ a−3, for matter,

ρr ∝ a−4, for radiation.(1.37)

4This concept will be introduced in the next section.

14

This is an easy way of concluding that, for this scenario, in a sufficiently distant past, the ultrarelativistic

species were dominant over the matter species and, when the scale factor became sufficiently large, the

matter fluids became the dominant contribution to the content of the Universe.

An evolution equation for the scale factor can be derived for a flat Universe introducing (1.36) into

(1.33) and solving for a(t):

a(t) ∝ t2

3(w+1) , (1.38)

which is valid whenever w 6= −1. Hence, the scale factor depends on time through a power-law solution

and, in particular, a(t) ∝ t2/3, for matter,

a(t) ∝ t1/2, for radiation.(1.39)

Equation (1.36) is only valid for the case where there is only one perfect fluid characterised by an

EoS parameter w, appearing in the Einstein equations (1.20). In principle, there will be multiple fluids

sourcing the cosmological equations. In that case, the total energy momentum tensor present in the

Einstein equations, has to account for an individual energy momentum tensor for each fluid:

Tµν = T(1)µν + T

(2)µν + ...+ T

(n)µν , (1.40)

where n is the number of species in the theory. In this case, the conservation equation (1.25) implies the

conservation of the total energy momentum tensor, i.e., of the sum of the energy momentum tensors for

each fluid:

∇µTµν = 0 =⇒ ∇µ(T (1)µν + T

(2)µν + ...+ T

(n)µν

)= 0. (1.41)

This means that we have no information on whether the energy momentum tensor for a single fluid

is individually conserved. For example, considering a theory with only two fluids, T (1)µν and T(2)µν :

∇µTµν = ∇µ(T (1)µν + T

(2)µν

)= 0 =⇒ ∇µT (1)µν = Qν and ∇µT (2)µν = −Qν , (1.42)

where Qν stands for the exchange of energy-momentum between both fluids. Indeed, whenever there

is an interaction present, the form of Qν should not be arbitrary, but should account for the physical

properties of each fluid and how that could lead to an interaction.

Taking into account the different fluids present in the Universe, equation (1.33) can be rewritten as

1 = Ωm(a) + Ωr(a) + Ωk(a), (1.43)

15

where

Ωk ≡ −k

H2a2(1.44)

and

Ωm,r ≡8πG

3H2ρm,r (1.45)

are the density parameters for curvature, matter fluids and ultrarelativistic fluids. This definitions, to-

gether with the previous analysis, suggests that we can rewrite (1.33) as

H2 = H20(Ωk0a

−2 + Ωm0a−3 + Ωr0a

−4) , (1.46)where Ωk0 , Ωm0 and Ωr0 correspond to the value of the parameters defined in (1.44) and (1.45) today,

and H0 also represents the value of the Hubble rate at present times.

1.4 Dark Energy

The discovery of the accelerated expansion of the Universe by the Supernova Cosmology Project [25]

and by the High-z Supernova Search Team [26] in 1998 made drastic changes regarding our knowledge

of the Universe. Currently, our best guide comes from the latest release of parameter estimates coming

from the Planck satellite observations [27, 28] of the cosmic microwave background (CMB). Based on

these, and other cosmological observations, it is very well established that the Universe is currently

undergoing a period of accelerated expansion.

One of the main challenges in cosmology concernes the determination of the composition of the

Universe. As shown above, ordinary matter such as dust or radiation cannot be the power source for ac-

celerating our Universe. Consequently, these results indicate that there seems to exist an unknown source

of energy/matter component, which happens to be the most abundant among the known constituents of

the Universe. Such an energy source would need to have an EoS characterised by an effective negative

pressure in order to explain the observations. It is impossible to build a macroscopic type of matter which

behaves in this manner considering only particles of the Standard Model, i.e., all types of matter we have

detected so far. Such an unknown component is generally classified under the broad heading of Dark

Energy (DE) [29]. These results could also be explained by a more general gravitational theory, studied

in the formalism of Modified Theories of Gravity [30]. DE attempts to explain the late-time acceleration

as a mass-energy component on the right side of the Einstein field equations.

Einstein himself attempted to solve this problem (1917, although it was not a problem at the time

[20]) while aiming for a static cosmological solution. Such a static situation still requires a component

16

with negative pressure, more precisely a combination with w = −1. Einstein chose to rule this rather

odd equation of state as non-physical and instead introduced his famous cosmological constant Λ to the

gravitational action. By doing so we get a set of modified Einstein field equations (1.20),

Rµν −1

2Rgµν + Λgµν = 8πGTµν . (1.47)

This static solution was found to be unstable. However, as was later discovered [31, 32], the Universe

is not static, but we can rely upon Einstein’s thinking to try to explain the accelerated expansion.

One of the main problems revolves around quantum field theory arguments, predicting the existence

of a vacuum energy density associated with quantum fields, which should make a contribution to the

effective EM tensor. A problem was found when such a vacuum contribution was theoretically estimated,

leading to a discrepency of about 120 orders of magnitude, when compared to the experimental value

of the cosmological constant [33]. This is the famous fine tuning problem of the cosmological constant

[34–38].

Inspired by Einstein’s formulation, the simplest model of dark energy is represented by the cosmo-

logical constant Λ, which is now related to a cosmological source with pΛ = −ρΛ, or wΛ = −1, and is

called ΛCDM model, where CDM stands for cold dark matter (and cold stands for non-relativistic).

Dark matter is an undetected entity which is needed at galactic and cosmological scales in order to

correctly fit the astronomical data [27, 28, 39, 40]. The measurements of the rotation curves of galaxies

are not in agreement with the theoretical predictions based on Newtonian mechanics. The observed

behaviour can only be explained if more mass is present. But this type of mass seems to neither emit nor

absorb radiation and therefore cannot be detected through electromagnetic interaction.

Assuming a ΛCDM model, the most recent astronomical observations seem to point for a cosmo-

logical constant model with ΩΛ0 = 0.6847 ± 0.0073, for its present density parameter [28]. In fact, by

combining the Planck data [28] with other observational data such as Type-Ia supernovae, it is found

w = −1.03 ± 0.03 for the EoS of dark energy, which is in agreement with the results for a cosmo-

logical constant. For the remainder content of the Universe it is found, Ωb0h2 = 0.02237 ± 0.00015

for baryonic content and ΩDM0h2 = 0.1200 ± 0.0012 for cold dark matter, where h = H0/(100 km

s−1 Mpc−1) is the usual way of introducing the observational uncertainty related to the Hubble param-

eter, which is constrained as H0 = 67.36 ± 0.54 km s−1 Mpc−1. The neutrino mass is also tightly

constrained as mν < 0.12 eV. These observations also suggest that the Universe is spatially flat, with

Ωk0 = 0.001± 0.002, which is a good argument to take k = 0 in the metric (1.28) and, consequently, in

(1.33).

In this scenario, at early times, the ultrarelativistic species prevailed over all the others and, when

the scale factor became sufficiently large, the matter fluids became the dominant contribution to the

17

content of the Universe, until very recently, when the dark energy component started to dominate. But,

as mentioned, in the standard ΛCDM model of cosmology, the Universe at the present day appears to

be extremely fine tuned, as this cosmological constant appears to have begun dominating the universal

energy at a very specific moment. The ΛCDM model is also known to present some conceptual problems,

such as the cosmic coincidence problem [41, 42]. In attempts to avoid these problems, the cosmological

constant is often generalised to a dynamical scalar field, whose time evolution could more naturally result

in the observed energy density today. The dynamics of the evolution of the Universe under the ΛCDM

model can be found, for example, in [43].

1.4.1 Quintessence Field and Tachyon Field

Scalar fields, describing scalar (spin 0) particles, are extremely important entities in modern physics.

Relevant examples of scalar fields are the recently detected Higgs field [44], responsible for the mech-

anism of providing mass to the particles of the Standard Model of Particle Physics, or the inflaton,

considered to be the scalar field that drives inflation [45]. Both these scalar fields play a critical role in

models of fundamental physics. The inflaton, in particular, gives rise to dynamics similar to dark energy,

since both have to be responsible for a period of accelerated expansion. Additionally, there is a precedent

of solving problems related to missing energy by hypothesising a new particle or field, as was the case

with the neutrino and dark matter (which still awaits proper detection). Scalar fields were first considered

in cosmology mainly in the context of a time varying cosmological constant [46–48]. For this reason, it

seems reasonable to assume that dark energy could also be described by a dynamical scalar field, varying

slowly along some potential V (φ), instead of the cosmological constant. This dynamical scalar field

should account for the missing energy contribution needed to preserve flatness in the Universe. This

mechanism is similar to slow-roll inflation in the early Universe [49], but the difference is that non-

relativistic matter (dark matter and baryons) cannot be ignored in order to properly discuss the dynamics

of dark energy. The DE equation of state varies dynamically, making these models distinguishable from

the standard ΛCDM model.

Another important motivation to consider a dynamical scalar field is the so-called “coincidence prob-

lem”, which concerns the initial conditions required to explain the value of the energy densities of matter

and dark energy today. In other words, it seems to be an incredible coincidence that currently the energy

densities of dark energy and dark matter are comparable in magnitude. For the case of the cosmological

constant, the only possible option is to extremely fine tune the ratio of energy densities at the end of

inflation. On the other hand, a dynamical scalar field could, in principle, couple to other forms of energy

(directly or simply through gravitational interaction), granting the possibility of a DE component which

naturally adjusts itself to reproduce the inferred energy density today. This can be achieved, for instance,

18

with a DE model with attractor-like solutions which reproduce the energy densities for a very wide range

of initial conditions.

Regarding this, scalar field based theories of dark energy are most commonly described by a canoni-

cal scalar field φ, the quintessence field [50], which is the simplest scalar field scenario, and is described

by a Lagrangian of the form:

Lquin = −1

2∂µφ∂

µφ− V (φ). (1.48)

Through a simple analogy with special relativity, it is immediately clear that this represents a consis-

tent generalisation of the classical Lagrangian of a non-relativistic particle,

L =1

2q̇2 − V (q), (1.49)

where q stands for generalised coordinates in the Hamiltonian formulation.

It goes without saying, that cosmological models which rely on a canonical scalar field in order to

account for the late time acceleration, fall under the cathegory of the so-called quintessence models.

These type of models were first introduced in [46, 51]. Some sample models with quintessence applica-

tions, such as quintessence driven inflation or dynamical quintessence models with different potentials

can be found in [52–58]. Throughout this work we will make use of some dynamical variables to de-

scribe quintessence models, which were first introduced in [59–63]. Different models can differ from

each other on the choice of the potential V (φ) in the Lagrangian (1.48). The simplest case is the one for

an exponential potential and was studied, for example, in [59, 64, 65] and for a power law potential in

[56, 60, 66, 67]. For other potentials see, for example [55, 62, 64, 68–78].

Analogously, one could look for a natural generalisation of the Lagrangian for a relativistic particle:

L = −m√

1− q̇2, (1.50)

with energy E = m/√

1− q̇2 and momentum p = mq̇/√

1− q̇2, related by E2 = p2 +m2.

This can be generalised for a scalar field and written as

Ltach = −V (φ)√

1 + ∂µφ∂µφ, (1.51)

where the field φ is termed the tachyon field.

This relativistic description allows for massless particles with a finite well-defined energy given by

the particles momentum: E2 = p2. This can be transposed to field theory by taking the equivalence

q(t) −→ φ and q̇2 −→ −gµν∂µφ∂νφ where φ is a scalar field which, by means of relativistic invariance,

could depend on both space and time. This also means that it is possible to treat the mass as a function

of the scalar field [79], through V (φ).

19

This approach takes us to the so called tachyon scalar field which is widely studied in the context

of string theory, due to its crucial role in the Dirac-Born-Infeld (DBI) action, used to describe the D-

brane action [80–90]. Tachyons were originally proposed as hypothetical particles which could travel

faster than light. This has turned them into a theoretical illness for a long time. As explained above,

in the context of Field Theory, this has acquired new meaning and tachyons are associated with the

description of quantum states with negative mass squared. This translates into a vacuum instability

which is represented by the negative sign in (1.51). The potential is initially at a local maximum, i.e., the

field is very carefully balanced at the top of the potential and any small perturbation will destabilise the

system and make the field roll towards the local minimum (e.g. the mexican hat potential). Through this

process the quanta become well-defined particles with a positive, real-valued mass.

It has already been showed that tachyons could play a useful role in cosmology [79, 91–110].

Tachyons as the source of dark energy have also been the focus of many studies [110–114]. In [115]

a complete dynamical study of a dark energy scenario in the presence of a tachyon field and a barotropic

perfect fluid for specific forms of the potential was performed. This was also done in [116] for a general

form of the potential. For the tachyon field, the simplest case corresponds to the inverse square potential,

which was studied with dynamical system techiques, for example, in [114, 115]. Other forms for the

potential were considered in [117, 118].

There has been great activity in this area, motivated by explaining the accelerated expansion of the

Universe without the problems of the cosmological constant. That is exactly what we aim for in this

work, by considering dark energy models described by the quintessence field and the tachyon field, with

the addition of allowing matter to naturally interact with dark energy through a conformal or disformal

transformation of the metric.

20

Chapter 2

Conformal and Disformal

Transformations

In this Chapter we will focus on the topic of transformations of the metric tensor. We start by giving

an introduction to the Lagrangian formalism of GR, based on the Einstein-Hilbert action, from where

we extract the previously introduced Einstein field equations. In the early Chapters we have argued the

possibility of having scalar fields present in our Universe. This will change the Lagrangian formalism of

the theory, which departs from standard GR and, as we will see in further detail, leads to the concept of

frames, i.e., theories which can be related by a metric transformation. In particular, when the action can

be cast into the usual Einstein-Hilbert action plus the terms related to the scalar field, the scalar field is

said to be minimally coupled to gravity and we speak of the Einstein frame. On the other hand, if it is the

matter sector which we write separately, shifting the effect of the scalar field into the gravitational sector,

then the field is said to couple minimally to matter and, in that case, we speak of the Jordan frame. In

this work, we discuss the so-called conformal transformations of the metric and its generalisation, the

disformal transformations. The use of conformal/disformal transformations has become conventional in

the literature regarding alternate theories of gravity based on GR. Typically, the transformation can be

accomplished through a function(s) of the fundamental scalar field, already present in the theory. We

will see that metric transformations are of paramount importance since they can be used to simplify the

form of the action and relate apparently different theories.

In the end of this Chapter we discuss the concept of interacting dark energy (IDE) theories and how

they can arise naturally from metric transformations. We also present a succinct review on the models of

IDE present in the literature.

21

2.1 Lagrangian Formalism of GR

In Classical Field Theory [119], the information regarding the dynamics of a given system can be

encapsulated in a mathematical function called the Lagrangian, L, which is a function of the generalised

coordinates, their time derivatives and time. In this formalism we can turn to Hamilton’s Principle, which

is formulated in terms of an action integral,

S =

∫ t2t1

L(qi(t), q̇i(t))dt, (2.1)

where qi represents the generalised coordinates. This principle states that the evolution of a system

described by N generalised coordinates, q1, ..., qN , between two specific states qi(t1) and qi(t2), at two

specified times t1 and t2, is characterised by the action being stationary under small variations in the

path: qi(t) 7−→ qi(t) + δqi(t), with δqi(t1) = δqi(t2) = 0. This condition can be written as

δS =

∫ t2t1

δLdt = 0. (2.2)

For this reason, this is also referred to as the “Principle of least action”. The variation of the action

leads to

δS =

∫ t2t1

[∂L

∂qiδqi +

∂L

∂q̇iδq̇i

]dt. (2.3)

It is possible to rewrite the second term in (2.3) as∫ t2t1

∂L

∂q̇iδq̇idt =

∫ t2t1

∂L

∂q̇iδ

(dqidt

)dt =

[∂L

∂q̇iδqi

]t2t1

−∫ t2t1

d

dt

∂L

∂q̇iδqidt. (2.4)

We find that the term[∂L∂q̇iδqi

]t2t1

vanishes, since we are considering δqi(t1) = δqi(t2) = 0 and so we

get

δS =

∫ t2t1

[∂L

∂qi− ddt

∂L

∂q̇i

]δqidt = 0. (2.5)

In order for δS = 0 to hold for any arbitrary variation δqi, the term inside the curved brackets in

(2.5) has to vanish. This yields the Euler-Lagrange equations:

∂L

∂qi− ddt

∂L

∂q̇i= 0 (2.6)

In field theory, the generalised coordinates qi(t) are replaced by a set of spacetime dependent fields

Φi(xµ), and S becomes a functional of these fields. The Lagrangian is now expressed as an integral

over space of a Lagrangian density L, which is a function of the fields Φi and their spacetime derivatives

∂µΦi,

22

L =

∫d3xL(Φi, ∂µΦi) =⇒ S =

∫d4xL(Φi, ∂µΦi). (2.7)

Analogously, it is possible to derive the Euler-Lagrange equations,

∂L∂Φi− ∂µ

(∂L

∂(∂µΦi)

)= 0. (2.8)

For a more detailed description see, for example, [10].

Variational principles are used to derive the equations describing the motion of particles and fields in

theoretical physics, and GR is not an exception. We need to define the Lagrangian density from which we

derive the Einstein field equations. Let us consider a four-dimensional spacetime manifoldM endowed

with a metric gµν , accommodating the Levi-Civita connection, i.e, ∇λgµν = 0. The Lagrangian density

is assumed to be a function of the metric and of its derivatives up to second order.

The relevant action should be a scalar. The simplest choice, from both the mathematical and physical

points of view, is the Ricci scalar curvature R, defined in (1.23), which includes only first and second

order derivatives of the metric tensor. Thus, we introduce the Einstein-Hilbert action plus a matter-energy

source, defined as

S = SEH + SM =

∫Ud4x√−g

M2p2R+

∫Ud4x√−gLM, (2.9)

where U is a compact region of the manifold, Mp ≡ 1/√

8πG,√−g is the invariant volume element

defined in terms of the determinant of the metric g, R is the Ricci scalar and LM is the Lagrangian density

for matter and energy.

The field equations are obtained from the variation of the action (2.9) through the principle of least

action, by imposing that the metric and its first derivatives are held fixed on the boundary ∂U ,

δS = δSEH + δSM = 0. (2.10)

We will focus first on the Einstein-Hilbert action:

δSEH =M2p2

∫Ud4x

[√−gδR+Rδ

√−g]. (2.11)

The term pro

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