Models of Disformally Coupled Dark Energy

Table of contents :
Acknowledgments......Page 3
Resumo......Page 5
Abstract......Page 9
List of Tables......Page 15
List of Figures......Page 17
1.1 Dynamical Systems......Page 19
1.1.1 Linear Stability Theory......Page 22
1.1.2 Bifurcations......Page 23
1.2 Basics of General Relativity......Page 27
1.3 Standard Model of Cosmology - The FLRW model......Page 30
1.4 Dark Energy......Page 34
1.4.1 Quintessence Field and Tachyon Field......Page 36
2 Conformal and Disformal Transformations......Page 39
2.1 Lagrangian Formalism of GR......Page 40
2.2 Conformal Transformations......Page 43
2.3 Disformal Transformations......Page 45
2.4 Einstein Frame and Jordan Frame......Page 48
2.5 Interacting Dark Energy......Page 50
3.1 The Model......Page 53
3.2 Background Cosmology......Page 56
3.3 Dynamical Equations......Page 58
3.4 Phase Space and Invariant Sets......Page 61
3.5.1 Fixed Points, Stability and Phenomenology......Page 62
3.5.3 Physical Phase Diagram......Page 67
3.6 Viable Cosmologies......Page 70
3.7 Effective Potential......Page 72
3.8 Summary......Page 75
4.1 The Model......Page 77
4.2 Background Cosmology......Page 80
4.3 Dynamical Equations......Page 82
4.4 Phase Space and Invariant Sets......Page 85
4.5.1 Fixed points, Stability and Phenomenology for a pressureless fluid......Page 86
4.5.2 Fixed points, Stability and Phenomenology for a relativistic fluid......Page 91
4.6 Conclusions......Page 94
5 Final Remarks......Page 97
Bibliography......Page 99
A Natural Units......Page 115
B Conformally Coupled Tachyonic Dark Energy - Linear Stability Matrix......Page 118
C Conformally Coupled Tachyonic Dark Energy - Eigenvalues of the Stability Matrix......Page 119
D Disformally Coupled Quintessence - Equation of Motion......Page 121
E Disformally Coupled Quintessence - Interaction term......Page 123
F Fixed Points for the Disformally Coupled System with no Kinetic Dependence......Page 125
G Disformally Coupled Quintessence - Eigenvalues of the Stability Matrix......Page 127

Citation preview

UNIVERSIDADE DE LISBOA ˆ FACULDADE DE CIENCIAS DEPARTAMENTO DE F´ISICA

Models of Disformally Coupled Dark Energy

Elsa Maria Campos Teixeira

Mestrado em F´ısica ˜ em Astrof´ısica e Cosmologia Especializac¸ao

˜ orientada por: Dissertac¸ao Doutor Nelson Nunes e Prof. Doutora Ana Nunes

2018

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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 Jos´e Pedro Mimoso and Francisco Lobo for the guidance, the encouragement and insightful comments. Also, to Inˆes, Rita and Isma for always being around, for the fruitful discussions 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 e´ o estudo do Universo, ou cosmos, como um todo. A Cosmologia padr˜ao surgiu com a formulac¸a˜ o da teoria da Relatividade Geral, em 1915, por Albert Einstein que, desde ent˜ao, se tem mostrado bem sucedida ao descrever a natureza do nosso Universo, tanto a pequenas como a largas escalas. Mostrou-se tamb´em capaz de fazer previs˜oes rigorosas que, s´o mais tarde, com o avanc¸o da tecnologia, puderam ser confirmadas, como e´ o caso das rec´em-detectadas ondas gravitacionais. Por isso, o modelo padr˜ao da Cosmologia baseia-se essencialmente na teoria da gravitac¸a˜ o de Einstein e nas suas implicac¸o˜ es cosmol´ogicas. No entanto, quando em 1998 se descobriu que o nosso Universo est´a a experienciar uma fase de expans˜ao acelerada, n˜ao parecia haver explicac¸a˜ o para este fen´omeno. Esta adic¸a˜ o ao nosso conhecimento acerca do Universo levou a` necessidade de considerar extens˜oes a` teoria da Gravitac¸a˜ o em vigor. Sabe-se que, todo o tipo de mat´eria conhecida e detectada at´e a` data, inclu´ıda no Modelo Padr˜ao da F´ısica de Part´ıculas, interage gravitacionalmente de forma atractiva (e nunca repulsiva), o que significa que, tendo apenas isso em conta, o Universo n˜ao se deveria estar a expandir. Neste sentido, um dos maiores desafios da Cosmologia diz respeito a` determinac¸a˜ o da composic¸a˜ o do Universo. Presentemente, uma das generalizac¸o˜ es mais bem aceites consiste em assumir que a expans˜ao acelerada do Universo e´ provocada pela presenc¸a de uma componente de mat´eria/energia, desconhecida at´e a` data, caracterizada por uma press˜ao eficaz negativa. Na realidade, atrav´es do ajuste dos dados observacionais, e´ poss´ıvel inferir que esta fonte desconhecida teria de ser a mais abundante entre todos os constituintes do Universo. A esta componente d´a-se o nome de Energia Escura. Postula-se ainda a existˆencia de um tipo de mat´eria, denominada Mat´eria Escura, necess´aria para ajustar correctamente os dados observacionais, que sugerem que deveria existir muito mais mat´eria do que aquela que se observa na realidade. Esta mat´eria n˜ao parece emitir nem absorver radiac¸a˜ o, pelo que n˜ao pode ser detectada por interacc¸a˜ o electromagn´etica. O pr´oprio Einstein, enquanto procurava uma soluc¸a˜ o para a sua teoria que permitisse descrever um Universo est´atico, verificou que seria necess´ario incluir uma contribuic¸a˜ o proveniente de uma componente de mat´eria com press˜ao negativa. Tratando-se de algo pouco usual, Einstein optou antes por adicionar a` sua teoria a famosa constante cosmol´ogica Λ. Mais tarde mostrou-se que esta soluc¸a˜ o apresenta instabilidades mas serviu de inspirac¸a˜ o a` s tentativas futuras de explicar a expans˜ao acelerada do Universo. Assim, o modelo mais simples de energia escura e´ o da constante cosmol´ogica Λ, que representa uma fonte cosmol´ogica com press˜ao pΛ = −ρΛ (onde p representa a press˜ao e ρ a densidade de energia que, por quest˜oes de consistˆencia, se assume sempre positiva). Com a adic¸a˜ o de uma componente de mat´eria escura obt´em-se o modelo padr˜ao da cosmologia actual: o modelo ΛCDM. Ainda assim, este modelo enfrenta alguns obst´aculos conceptuais e, por isso, considera-se extens˜oes em que a energia escura e´ descrita atrav´es de um campo escalar, cuja equac¸a˜ o de estado, p = wρ, varia dinamicamente, iii

sendo poss´ıvel reproduzir, com maior liberdade, a hist´oria da composic¸a˜ o 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 interpretac¸a˜ o f´ısica. Rapidamente se tornou natural considerar que a componente de energia escura possa interagir com a mat´eria convencional e com a mat´eria escura, dando origem a padr˜oes observacionais. Usualmente, e´ imposta uma func¸a˜ o de acoplamento ao n´ıvel das equac¸o˜ es do movimento. No entanto, esta interacc¸a˜ o pode emergir naturalmente atrav´es do pr´oprio campo escalar presente na teoria, que representa o papel de energia escura. Isto e´ poss´ıvel atrav´es de uma transformac¸a˜ o conforme/disforme do tensor da m´etrica, levando a uma interacc¸a˜ o descrita ao n´ıvel do Lagrangiano. As t´ecnicas desenvolvidas no contexto da teoria de Sistemas Dinˆamicos tˆem sido fulcrais para o desenvolvimento e interpretac¸a˜ o de modelos cosmol´ogicos. O seu uso permite n˜ao s´o descrever o nosso Universo no passado e no presente (onde os resultados podem ser comparados com os dados observacionais), mas tamb´em fazer previs˜oes (ou especulac¸o˜ es) acerca da sua evoluc¸a˜ o futura. Esta dissertac¸a˜ o baseia-se nessas mesmas t´ecnicas para desenvolver modelos cosmol´ogicos capazes de explicar a fase de expans˜ao acelerada do Universo que vivemos no presente, permitindo interacc¸o˜ es entre a energia escura e as restantes componentes de mat´eria/energia naturalmente presentes na teoria. Neste sentido, na explorac¸a˜ o dos modelos cosmol´ogicos propostos, conjugam-se dois pontos de vista complementares: faz-se uma an´alise dinˆamica baseada em princ´ıpios matem´aticos bem estabelecidos e uma an´alise das consequˆencias cosmol´ogicas, baseadas em hip´oteses motivadas e relevantes. No Cap´ıtulo 1 comec¸amos por fazer uma breve apresentac¸a˜ o dos conceitos utilizados nos Cap´ıtulos que se seguem. Primeiramente, e´ feita uma breve introduc¸a˜ o a` teoria matem´atica de Sistemas Dinˆamicos e a` s principais t´ecnicas utilizadas ao longo deste trabalho. De seguida, apresenta-se uma pequena introduc¸a˜ o a` teoria de Relatividade Geral e ao modelo padr˜ao da Cosmologia. Finalmente, discute-se o conceito de energia escura e as suas principais caracter´ısticas. E´ tamb´em feita a distinc¸a˜ o entre o campo escalar can´onico, ou quintessˆencia, e o campo escalar relativista, denominado taqui˜ao, no contexto de Teoria Quˆantica de Campo, com um paralelismo a` teoria cl´assica. Terminamos com uma breve revis˜ao das diferentes aplicac¸o˜ es cosmol´ogicas destes dois campos, presentes na literatura. No Cap´ıtulo 2 apresentamos o conceito de transformac¸o˜ es conformes/disformes e e´ feita uma an´alise do seu significado do ponto de vista matem´atico e f´ısico. De seguida mostramos como este tipo de transformac¸o˜ es pode ser usado para descrever modelos cosmol´ogicos em diferentes referenciais com interpretac¸o˜ es f´ısicas espec´ıficas. Neste sentido, somos naturalmente levados para a descric¸a˜ o onde se permite interacc¸o˜ es entre energia escura e as outras formas de mat´eria/energia presentes na teoria. Discute-se as principais consequˆencias cosmol´ogicas dessa mesma interacc¸a˜ o e apresenta-se uma breve revis˜ao dos diferentes tipos de acoplamentos considerados na literatura. Os acoplamentos proveiv

nientes de transformac¸o˜ es conformes/disformes destacam-se por surgirem naturalmente numa teoria que j´a cont´em um campo escalar, permitindo estudar um determinado modelo cosmol´ogico num referencial onde a interpretac¸a˜ o f´ısica e´ mais evidente e/ou conveniente. A principal novidade associada aos modelos acoplados e´ o aparecimento de pontos fixos, denominados scaling, que descrevem um Universo a evoluir para um estado onde as densidades de energia escura e mat´eria escalam uma com a outra. Assim, a existˆencia de acoplamentos pode ter um papel ben´efico, na medida em que estas soluc¸o˜ es podem ser usadas como forma de aliviar o problema da coincidˆencia cosmol´ogica, relacionado com a necessidade de escolher condic¸o˜ es iniciais espec´ıficas para o nosso Universo de forma a obter a configurac¸a˜ o que se observa hoje. No Cap´ıtulo 3 aplicamos as ideias descritas nos Cap´ıtulos anteriores a um modelo com um acoplamento conforme, onde o papel de energia escura e´ representado por um campo escalar taqui´onico e se admite um potencial quadr´atico inverso. Faz-se uma an´alise detalhada do ponto de vista dinˆamico e extraem-se as principais consequˆencias cosmol´ogicas. O estudo e´ feito em comparac¸a˜ o com o modelo desacoplado estudado anteriormente onde existe apenas um ponto fixo est´avel e capaz de descrever um Universo em expans˜ao acelerada, correspondente a um Universo a evoluir para um estado totalmente dominado por energia escura. Por outro lado, o modelo acoplado, admite soluc¸o˜ es de scaling, abrindo portas para novas configurac¸o˜ es. Com base na an´alise dinˆamica e no estudo de estabilidade dos pontos fixos encontrados, conclui-se que este modelo s´o e´ capaz de reproduzir a hist´oria da constituic¸a˜ o do Universo para condic¸o˜ es iniciais muito particulares. No Cap´ıtulo 4 implementamos um modelo com um acoplamento disforme, onde se toma um campo escalar can´onico para descrever a energia escura. Modelos de quintessˆencia com um acoplamento disforme j´a foram previamente considerados mas a an´alise existente na literatura e´ estendida, j´a que se considera que a func¸a˜ o disforme pode depender, n˜ao s´o do campo escalar, mas tamb´em das suas derivadas temporais e/ou espaciais. Ainda que o sistema seja complexo e extensivo do ponto de vista matem´atico, extraem-se as principais conclus˜oes cosmol´ogicas e, em particular, estuda-se em que medida a dependˆencia da func¸a˜ o disforme no termo cin´etico do campo escalar afecta a dinˆamica do sistema. Terminamos com o Cap´ıtulo 5, onde referimos as principais conclus˜oes do estudo realizado nesta dissertac¸a˜ o. Nomeadamente, discutimos as vantagens e desvantagens de considerar acoplamentos entre energia escura e as outras formas de mat´eria. No futuro seria importante constranger os dois modelos atrav´es dos dados observacionais dispon´ıveis. A an´alise detalhada da dinˆamica de cada sistema permite uma discuss˜ao das consequˆencias cosmol´ogica mais consistente, num tema t˜ao pertinente e instigante como a hist´oria da composic¸a˜ o do Universo e a sua presente expans˜ao acelerada.

Palavras-chave:

Energia Escura, Sistemas Dinˆamicos em Cosmologia, Transformac¸o˜ es Con-

formes/Disformes, Quintessˆencia Acoplada, Taqui˜ao v

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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 cosmological 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 Cosmology: 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 conventional 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 conformal/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 already 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 conformally 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 describing 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 extensive 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 Transformations, Coupled Quintessence, Tachyon ix

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Contents Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

3

xv

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

18

Quintessence Field and Tachyon Field . . . . . . . . . . . . . . . . . . . . . . .

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

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

4

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

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

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.6 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 . . . .

3.2

Fixed points and corresponding existence conditions and cosmological parameters for

42

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 coupled to a non-relativistic fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

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

4.3

74

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 . . . . . . .

F.1

70

Fixed points for disformal quintessence coupled to a relativistic fluid with a linear dependence of the disformal function on the kinetic term . . . . . . . . . . . . . . . . . .

4.4

69

74

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

xiv

List of Figures 1.1

Saddle node and transcritical bifurcation diagrams . . . . . . . . . . . . . . . . . . . . .

3.1

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

8

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 according 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

4.2

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

72

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

73

xv

xvi

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, dx1 = x˙ 1 = f1 (t, x1 , ..., xn ), dt .. .

(1.1)

dxn = x˙ n = fn (t, x1 , ..., xn ), dt 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 x˙ = 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, x˙ = 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. LotkaVolterra models, where the state space corresponds to the possible number of elements in each pop1

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 ψ(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 x˙ = 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 ˙ ψ(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 x˙ = 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 x˙ = 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 x˙ = 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 x˙ = 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 x˙ = 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 x˙ = 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 x˙ = 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 n X ∂fi 1 X ∂ 2 fi (x∗ )uj + (x∗ )uj uk + ..., ∂xj 2! ∂xj ∂xk j=1

(1.2)

j,k=1

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 X ∂fi (x∗ )uj . ∂xj

(1.3)

j=1

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=

h

∂fi ∂xj

i



∂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 1 ∂xj .

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. 1

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

4

Theorem 1.1.2. Let x∗ be a fixed point of the system x˙ = 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 of M(x∗ ) have negative real parts then the fixed point is asymptotically stable and is said to be an attractor. • If all of the eigenvalues of M(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 of M(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 of M(x∗ ) have real part equal to zero. This leads us to the definition of hyperbolic fixed point: Definition 1.1.7 (Hyperbolic fixed point). Let x∗ be a fixed point of the system x˙ = 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 of M(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. 2

Note 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: x˙ = 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 determined by the sign of the eigenvalues of the stability matrix in the linear approximation. Indeed, when F(x∗ ; λ∗ ) = 0 and M(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 condition 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 = f (x; λ), dt

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 6

∂f ∂x (x∗ ; λ∗ )

= 0.

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 bifurcation 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∗ + fλ

dλ = 0. dx∗

(1.7)

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

dλ dx∗

= 0.

Example 1.1.1 (Saddle-node bifurcation). Consider the one-dimensional quadratic equation, dy = f (y; λ) = λ − y 2 , dt

(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 =

√ y∗2 = − λ

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

df dy

= −2y vanishes at y = 0,

where the fixed points collide giving rise to a single non-hyperbolic fixed point. Besides, √ √ df −2y∗1 = −2 λ and dy (y∗2 ; λ) = −2y∗2 = 2 λ.

df dy (y∗1 ; 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,

df dλ

= 1 6= 0 and equation (1.7) implies

dλ dy∗

= 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

y*

n*

λ

λ

(a) Saddle node bifurcation

(b) Transcritical bifurcation

Figure 1.1: Bifurcation diagrams for the saddle node bifurcation described in Example 1.1.1 (left) and the transcritical bifurcation described in Example 1.1.2 (right). A solid curve is used to represent a family of stable fixed points whereas a dashed curve is used to highlight a family of unstable fixed points. The vertical lines represent the flow generated by each system along the vertical direction. dn = f (n; λ) = λn − n2 , dt

(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

df dn

= 0 is when n = 0 and λ = 0, highlighting the

non-hyperbolic character of this fixed point. Actually, we have, df (n∗ , λ) = λ, dn 1

df (n∗ , λ) = −λ. dn 2

By linear stability theory, the fixed point n∗1 is stable if stable if

df dn (n∗2 , λ)

df dn (n∗1 , λ)

(1.12) < 0 and the fixed point n∗2 is

< 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),

dλ dn∗

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 gravitation. 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, g µν , which, given g 6= 0 (i.e., gµν invertible), is fully determined by the condition: gµα g αν = δµν , where the symmetry of gµν implies symmetry on the inverse, g µν . 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: X µ = g µν Xν .

(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): X µ = g µν Xν =

3 X ν=0

9

g µν Xν .

(1.14)

The separation between two points is defined according to the line element, ds, ds2 = gµν dxµ dxν ,

(1.15)

and is given by Z

B

`AB =

ds,

(1.16)

A

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: ∇µ Xα = ∂µ Xα − Γβµα Xβ ,

∇µ X α = ∂µ X α + Γαµβ X β ,

(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, ∇α gµν = 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 1 Γλµν = g λδ (∂ν gδµ + ∂µ gδν − ∂δ gµν ), 2

(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): 8πG 1 Gµν = Rµν − gµν R = 4 Tµν , 2 c

(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 ρ tensor, defined in terms of the Riemann tensor Rµσν , as follows: ρ Rµσν = ∂σ Γρνµ − ∂ν Γρσµ + Γρσλ Γλνµ − Γρνλ Γλσµ ,

10

(1.21)

λ Rµν = Rµλν ,

(1.22)

R = g µν Rµν .

(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, ρ ρ ρ ∇λ Rµσν + ∇ν Rµλσ + ∇σ Rµνλ = 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: ∇µ Gµν = 0 =⇒ ∇µ Tµν = 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 µ ν d2 xα α dx dx + Γ = 0, µν ds2 ds ds

(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

d2 xα du2

µ

+ Γαµν dx du

dxν 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 gµν V µ V 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 dτ 2 = 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 d2 xα /dτ 2 = 0. For non-relativistic speeds dτ /dt ' 1 and the geodesic equation yields d2 xi /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 cosmology 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 assumptions 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 Principle, 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, spatially 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-Lemaˆıtre-Robertson-Walker metric [21–23] and has the following form:   dr2 2 2 2 2 2 2 2 2 ds = −dt + a (t) + r dθ + r sin θ , dϕ , 1 − kr2

(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)δij dxi dxj ,

(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)uµ uν − pgµν ,

(1.30)

where uµ is the fluid’s four-velocity defined as dxµ uµ = √ , −ds2

(1.31)

which satisfies uµ uµ = −1 and, for a comoving observer, is given by uµ = (−1, 0, 0, 0). Given this, and taking the metric in (1.29), Tνµ is a purely diagonal tensor:   −ρ 0 0 0    0 p 0 0 µ  Tν =     0 0 p 0 0

(1.32)

0 0 p

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: 3 The 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

 2 a˙ 8πG k H = = ρ − 2, a 3 a 2

where H ≡

a˙ a

(1.33)

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: a ¨ 4πG =− (ρ + 3p), a 3

(1.34)

which is known as the Raychaudhuri equation. This last equation is of great value as it gives a straightforward condition on the parameters in order to draw a distinction between a universe with an increasing expansion rate, if a ¨ > 0, or decreasing, if a ¨ < 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 physically 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: ρ˙ + 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,  ρ ∝ a−4 , r 4

for radiation.

This concept will be introduced in the next section.

14

(1.37)

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): 2

a(t) ∝ t 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: (1) (2) (n) Tµν = Tµν + Tµν + ... + Tµν ,

(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:   (1) (2) (n) ∇µ Tµν = 0 =⇒ ∇µ Tµν + Tµν + ... + Tµν = 0.

(1.41)

This means that we have no information on whether the energy momentum tensor for a single fluid (1)

(2)

is individually conserved. For example, considering a theory with only two fluids, Tµν and Tµν :   (1) (2) (1) (2) ∇µ Tµν = ∇µ Tµν + Tµν = 0 =⇒ ∇µ Tµν = Qν and ∇µ Tµν = −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), 15

(1.43)

where k H 2 a2

(1.44)

8πG ρm,r 3H 2

(1.45)

Ωk ≡ − and Ωm,r ≡

are the density parameters for curvature, matter fluids and ultrarelativistic fluids. This definitions, together with the previous analysis, suggests that we can rewrite (1.33) as
H 2 = H02 Ωk0 a−2 + Ωm0 a−3 + Ωr0 a−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 accelerating 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), 1 Rµν − Rgµν + Λgµν = 8πGTµν . 2

(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 cosmological 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 cosmological 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 cosmological constant. For the remainder content of the Universe it is found, Ωb0 h2 = 0.02237 ± 0.00015 for baryonic content and ΩDM 0 h2 = 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 parameter, 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 mechanism 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 nonrelativistic 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 problem”, 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 canonical scalar field φ, the quintessence field [50], which is the simplest scalar field scenario, and is described by a Lagrangian of the form: 1 Lquin = − ∂µ φ∂ µ φ − V (φ). 2

(1.48)

Through a simple analogy with special relativity, it is immediately clear that this represents a consistent generalisation of the classical Lagrangian of a non-relativistic particle, 1 L = q˙2 − V (q), 2

(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 applications, 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 describe 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 with energy E = m/

p 1 − q˙2 ,

(1.50)

p p 1 − q˙2 and momentum p = mq/ ˙ 1 − q˙2 , related by E 2 = p2 + m2 .

This can be generalised for a scalar field and written as Ltach = −V (φ)

p 1 + ∂µ φ∂ µ φ,

(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: E 2 = p2 . This can be transposed to field theory by taking the equivalence q(t) −→ φ and q˙2 −→ −g µν ∂µ φ∂ν φ 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 Dbrane 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, Z

t2

S=

L(qi (t), q˙i (t))dt,

(2.1)

t1

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 Z t2 δLdt = 0. δS =

(2.2)

t1

For this reason, this is also referred to as the “Principle of least action”. The variation of the action leads to t2

Z δS =

t1



 ∂L ∂L δqi + δ q˙i dt. ∂qi ∂ q˙i

It is possible to rewrite the second term in (2.3) as    t2 Z t2 Z t2 Z t2 dqi ∂L ∂L ∂L d ∂L dt = δ q˙i dt = δ δqi δqi dt. − dt ∂ q˙i t1 ∂ q˙i t1 ∂ q˙i t1 dt ∂ q˙i t1 We find that the term

h

it2 ∂L δq i ∂ q˙i t1

(2.3)

(2.4)

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

get Z

t2

δS = t1



 d ∂L ∂L − δqi dt = 0. ∂qi dt ∂ q˙i

(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: d ∂L ∂L − =0 ∂qi dt ∂ q˙i

(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 ∂µ Φi , 22

Z L=

3

i

Z

i

d xL(Φ , ∂µ Φ ) =⇒ S =

d4 xL(Φi , ∂µ Φi ).

(2.7)

Analogously, it is possible to derive the Euler-Lagrange equations, ∂L − ∂µ ∂Φi



∂L ∂(∂µ Φi )

 = 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 manifold M endowed with a metric gµν , accommodating the Levi-Civita connection, i.e, ∇λ gµν = 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 √ Mp2 d4 x −g R+ 2 U

Z S = SEH + SM =

Z

√ d4 x −gLM ,

(2.9)

U

√ √ 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 =

Mp2 2

Z

d4 x

√

√  −gδR + Rδ −g .

(2.11)

U

The term proportional to δR in (2.11) can be expanded, using the definition of the Ricci scalar: δR = Rµν δg µν + g µν δRµν .

(2.12)

The second term in (2.12) disappears when integrated, according to Stokes’ Theorem [14]: Z

√ d4 xg µν −gδRµν = 0.

U

23

(2.13)

To manipulate the term proportional to

√

−g in (2.11) we can rely on the fact that, for any square

matrix M with determinant det(M ) 6= 0: ln(det(M )) = Tr(ln(M )) ⇒

1 δ det(M ) = Tr(M −1 δM ). det(M )

(2.14)

Taking M = g µν and det(M ) = g: √ 1 1√ −ggµν δg µν . δ −g = − √ δg = − 2 −g 2

(2.15)

Finally, δSEH

Mp2 = 2

√

  1 d x −g Rµν − Rgµν δg µν , 2 U

Z

4

for the Einstein-Hilbert action. For the matter action we find   Z Z   √ √ √ δLM 1 µν − g L δSM = d4 x LM δ −g + −gδLM = d4 x −g µν M δg , δg µν 2 where we relate the term between square brackets with the energy-momentum tensor as   √ −2 δ( −gLM ) δLM 1 (m) Tµν = √ = −2 − gµν LM . −g δg µν δg µν 2

(2.16)

(2.17)

(2.18)

Hence, using the principle of least action, δS = δSEH + δSM = 0 ⇒ δSEH = −δSM ,

(2.19)

from where we derive the Einstein field equations (in the absence of a cosmological constant) 1 (m) Rµν − Rgµν = κ2 Tµν , 2 with κ ≡ Mp−1 ≡

√

(2.20)

8πG, as Euler-Lagrange equations of the Einstein-Hilbert action plus a matter

source. As a final remark, we can take the trace of equation (2.20): R = −κ2 T (m) ,

(2.21)

where T stands for the trace of the energy-momentum tensor. Taking this into account, we can rewrite (2.20) as Rµν

  1 (m) (m) . = κ Tµν − gµν T 2 2

This alternative representation of the Einstein field equations is entirely equivalent to (2.20). 24

(2.22)

2.2

Conformal Transformations

Let us consider a Riemannian manifold (M, gαβ ) of a given dimension n. The map gαβ 7−→ geαβ = Ω2 (x)gαβ ,

(2.23)

is conventionally called a conformal transformation. We will use a tilde to denote quantities in the transformed frame. In this definition, the conformal factor Ω(x) is some non-vanishing regular spacetime function. This map is known to leave the angle between any two vectors of the spacetime invariant. For the purpose of showing this, let xa and y b , with a, b = 1, ..., n, be any two vectors on the manifold. By definition, the angle between the two vectors is given through the dot product by: (x · y) gab xa y b cos(θ) ≡ p . =p (gcd xc xd )(gef y e y f ) (x · x)(y · y)

(2.24)

Applying the conformal transformation (2.23) this becomes geab xa y b Ω2 (x)gab xa y b =p = cos(θ), cos(θ) 7−→ p (e gcd xc xd )(e gef y e y f ) (Ω2 (x)gcd xc xd )(Ω2 (x)gef y e y f )

(2.25)

which proves angle preservation. On the other hand, the distance between two given points on the manifold is changed1 . This effect can be seen by how the transformation on the metric affects the line element (1.15): ds2 7−→ de s2 = geαβ xα xβ = Ω2 gαβ xα xβ = Ω2 ds2 .

(2.26)

Therefore, given any two points A and B on the manifold, the distance between them `AB , as defined in (1.16), after the conformal transformation, is given by Z B Z e `AB 7−→ `AB = de s= A

B

Ωds 6= `AB .

(2.27)

A

This means that a conformal transformation will affect the length of time-like and space-like intervals and the norm of time-like and space-like vectors while leaving the light cones unchanged. As a consequence, the spacetimes (M, gµν ) and (M, geµν ) have the same causal structure (as defined in (1.27)). From (2.23) we have g αβ 7−→ geαβ = Ω−2 (x)g αβ

(2.28)

for the inverse metric and 1 Note that, when performing a conformal transformation of the metric, the spacetime coordinates xµ do not change relative to the manifold, since they are fixed on the manifold. It is the structure of the manifold (through the metric) which changes.

25

g ≡ det(gαβ ) 7−→ ge ≡ det(e gαβ ) = Ω2n (x)g,

(2.29)

for the determinant of the metric, where n is the dimension of the spacetime manifold M (we always take n = 4). In general, tensorial quantities are not invariant under conformal transformations, suggesting that, in principle, neither will the tensorial equations. Invariance under conformal transformations is a very exceptional property. The energy-momentum tensor (2.18), defined in terms of the matter action in (2.9), transforms according to (m) (m) Teµν = Ω−2 Tµν ,

(2.30)

Teµν (m) = Ω−4 Tµν (m) ,

(2.31)

µν µν Te(m) = Ω−6 T(m) ,

(2.32)

Te(m) = Ω−4 T (m) .

(2.33)

and

(m)

It can be shown that the conservation relation (1.25) for a symmetric stress-energy tensor Tµν is µ (m)

only found to be conformally invariant whenever the trace T = Tµ

vanishes2 [13]:

(m) e ν (ln Ω). e µ Te(m) = −Te(m) ∇ ∇µ Tµν = 0 7−→ ∇ µν

(2.34)

From (2.33) it is clear that the trace of the energy-momentum tensor in the transformed frame can only vanish if it also vanishes in the original frame. Whenever T (m) 6= 0, equation (2.34) describes an ingoing/outgoing energy flow, where its rate depends on the choice for the conformal factor Ω. In other words, in the transformed frame description there is a direct coupling between matter and the geometric factor Ω. As a consequence, the free fall motion of particles has to account for an additional term which e µ Ω (this is sometimes identified as a fifth force acting on can be thought of as a force proportional to ∇ all massive particles). Let us consider that this energy momentum tensor can be described as a perfect fluid, as defined in (1.30). It transforms as (m) (m) Tµν 7−→ Teµν = (e p + ρe) u eµ u eν + pe geµν ,

(2.35)

2 Note that, when performing a conformal transformation of the metric, there is a redefinition of the Christoffel symbols and, e consequently, a redefinition of the covariant derivative: ∇ 7−→ ∇.

26

where, in the same manner as before, the four-velocity in the transformed frame has to satisfy u eµ u eµ = −1. This automatically leads to uµ 7−→ u eµ = Ωuµ and uµ 7−→ u eµ = Ω−1 uµ .

(2.36)

Comparing (2.30) and (2.35), it is possible to infer that p 7−→ pe = Ω−4 p and ρ 7−→ ρe = Ω−4 ρ.

(2.37)

Taking this into account, we find for the equation of state parameter in the transformed frame, defined according to pe = we e ρ, w e = w,

(2.38)

which means that the equation of state parameter for barotropic fluids is invariant under conformal transformations. The use of conformal transformation techniques has become conventional in the literature regarding modified theories of gravity (alternative theories based in GR). Typically, the transformation to the conformal frame is accomplished through a fundamental scalar field which was already naturally present in the theory. To do so, the dependence of the conformal factor Ω(x) on the spacetime points x is hidden in a functional dependence on the scalar field φ(x): Ω(x) = Ω[φ(x)].

(2.39)

Naturally, in this case, the conformal transformation is also connected to a redefinition of the scalar field itself e φ 7−→ φ.

(2.40)

Usually one of the metrics is taken to describe the gravitational part of the theory whereas the other is responsible for laying out the geometry in which matter plays out its dynamics.

2.3

Disformal Transformations

However, the conformal transformation is just the simplest way to relate two geometries. We could instead assume that the transformation depends not only on the scalar field itself but also on its first order partial derivatives. This leads to the so-called disformal transformation gαβ 7−→ g¯αβ = Ω2 (φ)gαβ + Γ(φ)∂α φ∂β φ , 27

(2.41)

where Γ is the disformal factor. Disformal transformations can be seen as a generalisation of conformal transformations. Note that these cannot be interpreted as a simple field redefinition, since they also include derivatives of the scalar field. Thus, the treatment should be more delicate. They do not represent a change in coordinates, but a local change in the geometry instead. In opposition to the conformal case, the disformal transformation could change the causal structure of the spacetime. A disformal transformation can be interpreted as a stretching (or a compression) of the metric in a specific direction defined according to the gradient of the scalar field. Overall, the disformal transformation, in opposition to the conformal one, deforms the spacetime in a preferred direction, resulting in a distortion of both angles and lengths. This formalism was first brought to attention in the cosmological community by Bekenstein [120] while looking for a general way to couple matter to the gravitational metric. But disformal transformations were only brought to the spotlight [121] when it was shown that they could be linked to Horndeski theories [122]. Following this, new scalar-tensor theories of gravity were derived such as beyond Horndeski or GLPV theory [123–126]. Currently, disformal transformations in cosmology have become conventional in research fields, such as brane world models [127–129], massive gravity theories [130], MOND theories [131, 132], extensions of dark matter [133, 134] and chameleon theories [135]. They are also found in theories in which Lorentz invariance is broken spontaneously on a non-trivial background [136] and have been used to study inflation [137, 138] in the early Universe, dark energy [139–142], atomic physics [143] and mimetic gravity [144]. They have also been studied in the language of differential forms [145, 146] and used in generalised vector field theories [147–149]. Disformally coupled scalar fields were also applied to varying speed of light cosmologies [150–152]. The disformal transformation in (2.41) can be generalised as gαβ 7−→ g¯αβ = Ω2 (φ)gαβ + Γ(φ, X)∂α φ∂β φ,

(2.42)

where X = − 12 ∂µ φ∂ µ φ is the kinetic energy associated with the scalar field. In this work, for simplicity, we will restrict ourselves to the case where the conformal factor Ω depends only on scalar field φ, whereas the disformal factor Γ is allowed to depend both on φ and its associated kinetic term X. It has been shown that, in some theories where Ω also depends on X, Ostrogradski’s instibilities [153] may arise [121, 124, 154]. In this section we present the novel results of such a generalisation, developed in the context of this work. We derive g

αβ

αβ

7−→ g¯

−2

=Ω

  Γ∂ α φ∂ β φ αβ g − 2 , Ω − 2ΓX

for the inverse metric and 28

(2.43)

g 7−→ g¯ = Ω2n−2 (Ω2 − 2ΓX) g

(2.44)

for the determinant of the transformed metric, where we have omitted the φ and X dependences for simplicity and n is the dimension of the spacetime manifold M (we always take n = 4). Clearly, if we set Γ = 0 in (2.42), (2.43) and (2.44) we recover the conformal case. The energy-momentum tensor (2.18), defined in terms of the matter action in (2.9), no longer transforms linearly and can be computed as follows:   √ p gρσ ¯ρσ −¯ g δ¯ 1 αβ ρσ 3 2 α β α β 2 T(m) = √ T = Ω Ω − 2ΓX Ω δρ δσ − Γ,X ∂ φ∂ φ∂ρ φ∂σ φ T¯(m) , −g δgαβ (m) 2 (m)

Tαβ

" # √ 1 2 2 p Ω Γ + Γ g ρσ ¯(m) −¯ g δ¯ ,X (m) =√ T = Ω Ω2 − 2ΓX δαρ δβσ + 2 2 ∂α φ∂β φ∂ ρ φ∂ σ φ T¯ρσ , −g δg αβ ρσ (Ω − 2ΓX)2

(2.45)

(2.46)

and T (m) = Ω3

p   ρσ Ω2 − 2ΓX Ω2 gρσ + Γ,X X∂ρ φ∂σφ T¯(m) ,

(2.47)

where a comma denotes partial derivatives. It is clear that T (m) = 0 is no longer a condition when it comes to disformal invariance. Unlike conformal transformations, disformal transformations have non-trivial effects on radiation-like fluids, allowing for modifications in the behaviour of photons [155– 158]. As this transformation is very general, these models can in principle be very interesting from the phenomenological point of view. Now, assuming a perfect fluid description for the energy momentum tensor, (1.30), we get: (m) (m) Tµν 7−→ T¯µν = (¯ p + ρ¯)¯ uµ u ¯ν + p¯g¯µν ,

(2.48)

where u ¯µ is the four-velocity in the transformed frame, defined in (1.31) in terms of the line element which, according to (1.29), in Cartesian coordinates, transforms accordingly:
¯ 2 = g¯µν dxµ dxν = − Ω2 − 2ΓX dt2 + Ω2 a2 (t)δij dxi dxj . ds2 7−→ ds

(2.49)

This background metric choice guarantees that the fluids are still homogeneous, meaning that each element follows a geodesic prescribed by the metric tensor. In this sense, the four-velocity can still be computed directly from the metric. Taking into account the imposition u ¯µ u ¯µ = −1, computation of the four-velocity leads to: p dxµ uµ uµ 7−→ u ¯µ = ¯ = √ , and uµ 7−→ u ¯µ = g¯µν uν = Ω2 − 2ΓXuµ . |ds| Ω2 − 2ΓX

(2.50)

Following this, comparing the time-time components of (2.46) and (2.48), it is possible to infer that the energy density transforms as 29

√

ρ 7−→ ρ¯ =

Ω2 − 2ΓX ρ. Ω3 (Ω2 − 2Γ,X X 2 )

(2.51)

and, accordingly, comparison of the space-space components yields the transformation for the pressure p 7−→ p¯ =

Ω3

√

p . Ω2 − 2ΓX

(2.52)

Finally we find, for the equation of state parameter in the transformed frame, defined according to p¯ = w ¯ ρ¯, w 7−→ w ¯=

Ω2 − 2Γ,X X 2 w, Ω2 − 2ΓX

(2.53)

which means that the equation of state parameter for barotropic fluids is not disformally invariant. When performing a disformal transformation between frames, the mapping (2.53) cannot be disregarded.

2.4

Einstein Frame and Jordan Frame

The success showed by special relativity on providing a consistent description of the known elementary particle phenomena is usually taken as evidence to imply that spacetime is fully described by a Riemannian geometry. However, the physics of the Universe could be more complex: a coherent theory of gravitation could naturally call for a transformation between two geometries for its description. A conformally flat scalar theory of gravity was remarkably suggested for the first time in Nordstr¨om’s 1912 gravitational theory [159], prior to Einstein’s formulation of GR. In this theory, such as in theories like Jordan-Brans-Dicke [160], string theories [161], the variable mass theory [162], and many others, two geometries appear which can be conformally/disformally related. A safe way to proceed in order to avoid immediate conflict with the tests that GR has passed, is to invoke a Riemannian metric gµν , use it to build the Einstein-Hilbert action for the gravitational dynamics, and drift away from standard GR by prescribing a relation between gµν and the physical geometry on which matter is considered to be propagating. In other words, one simple way to introduce a non-trivial coupling between a scalar field and matter, is to envision that matter particles follow geodesics defined with respect to a transformed metric geµν which is related to the gravitational metric gµν by a conformal transformation. After performing a conformal transformation, the term “conformal frame” is conventionally used to discriminate the new transformed metric from the original. Of course there are infinite possible choices for the conformal factor, leading to infinite possible conformal frames3 . Among these, there are two which are often referred to due to their interpretations: 3 The word “frame” can sometimes be misleading. In this context we refer to a frame as a specific set of variables, in this case the metric and the scalar field, rather than a specific spacetime reference frame.

30

• The Jordan frame is the one in which the energy-momentum tensor is covariantly conserved, e µ Te(m) = 0, ∇ µν

(2.54)

e This corresponds to the scenario and, in this context, is defined by the new set of variables (e gµν , φ). in which test particles follow geodesics of the spacetime metric geµν . In this frame, the gravitational field is defined by the metric tensor and the scalar field present in the theory, and matter is said to couple minimally to the scalar field. For example, the Jordan-Brans-Dicke theory [160] is most usually formulated in the Jordan frame. • The Einstein frame is the conformal frame in which the field equations of the theory take the form of the Einstein equations and is defined by the set of variables (gµν , φ). This leads to second order field equations but it comes at a price: the energy-momentum tensor of the matter fields is not automatically covariantly conserved, due to the coupling of the matter fields with the scalar field, and test particles do not necessarily follow geodesics of the metric geµν . This is why it is usually said that, in the Einstein frame, it is the geometry (gµν ) which is minimally coupled to the field, inducing a non-minimal coupling between matter and the scalar field. In this frame the gravitational field is purely described in terms of the metric tensor gµν , and the scalar field φ acts as an external “matter field”, an imprint of its gravitational role in the Jordan frame. In broader terms, the frame where the scalar field is minimally coupled to matter is known as the Jordan frame, whereas the frame where it couples minimally to gravity is called the Einstein frame. Note that, unlike the Jordan frame, the Einstein frame can only be defined for some theories. When this is the case, by choosing the parameters of the transformation appropriately, it is possible to isolate an EinsteinHilbert term in the action and write the remaining terms as an effective scalar field propagating in the spacetime. There is a discussion centred upon whether the frames are found to be equivalent and, morover, if they indeed are equivalent, on whether this is a physical equivalence4 , in the sense that observables can be calculated in either frame, or purely mathematical, in which case the physical observables should be calculated in the “physical frame”. This leads to the issue of which conformal frame is the “physical frame”, which has been an ongoing topic of discussion [163, 164]. With the use of conformal transformations, one is able to map non-standard theories of GR, characterised by the Einstein-Hilbert Lagrangian, plus a scalar field φ, which is minimally coupled to the geometry in the Einstein frame. This points to the fact that the problem concerning the physical nature of 4 By “physical frame” we refer to a set of variables which, in principle, are measurable and satisfy every general requirement of a consistent classical field theory, such as the emergence of positive-definite energy densities.

31

the fields in each frame should be addressed prior to any other discussion regarding the physical nature of the equivalence between the frames. Recent works have shown the absolute physical equivalence between the Jordan and Einstein conformal frames [165], arguing that they are both equally legitimate as physical descriptions. Often, the metric in the Einstein frame is referred to as the gravitational metric whereas the metric in the Jordan frame is called the matter metric. This Einstein-frame and Jordan-frame distinction is also valid for disformally related metrics.

2.5

Interacting Dark Energy

The interest in the possibliliy of having dark energy coupled to the matter fluids present in the theory, and its corresponding cosmological applications, arose naturally. Motivation for considering models of interacting dark energy has come from the fact that these could potentially address the cosmological constant and coincidence problems [47, 166] and, additionallly, affect structure formation predictions in unprecedent ways, providing a way to alleviate tensions between the standard non-interacting dark energy models and observations [167]. However, this may come as a burden, as these interacting models often carry instabilities at the perturbation level. Another catalyst for the emergence of interacting dark energy was the inflation paradigm, first discussed in [168, 169]. By the late 1990’s this had led to an outpouring of scalar field inflationary models in cosmology [170], including many interacting models, mainly motivated by particle physics, e.g. [59, 171]. This meant that many of the theoretical and analytical techniques, needed to study and deal with these models, were all developed and readily applicable. As described in Section 1.3, the imposition of the Bianchi identities (1.41), i.e., that the total energymomentum tensor must be conserved, in the presence of an interaction between the mater sector and dark energy, leads to (m) ∇µ Tµν = −Qν ,

(DE) and ∇µ Tµν = Qν ,

(2.55)

where the superscript (m) stands for matter and (DE) for dark energy and Qν is the interaction vector which defines the strength of the coupling. (m)

It seems reasonable to assume that the matter energy momentum tensor Tµν accounts for the whole non-relativistic matter sector and is well described by a perfect fluid with wm = p/ρ = 0. Dark energy is allowed to interact with both dark matter and baryonic matter on cosmological scales, as long as the observational constraints are met. Although the couplings to baryonic matter are more stringent (from Solar System bounds) [172], this contribution can be ignored as a first approximation. This is taken into account in most of the works present in the literature, since dark matter stands for the majority of the pressureless matter sourcing the RHS of the Einstein equations. The baryonic constraints can also be 32

avoided by means of screening mechanisms [30, 173]. We will only consider interactions between DE and radiation for disformal couplings since radiation was found to be conformally invariant (see Section 2.2). For what concerns the work done so far, we will focus mainly on the interaction between the radiation and matter fluids with dark energy separately, rather than accounting for three fluids in (2.55). This leads to an equation similar to (2.55) for the radiation component (with (m) ↔ (r)). From a dynamical system point of view, the advantage of taking intreacting dark energy models is the possibility of introducing new scaling solutions (accelerating ones, hopefully, with weff < −1/3) which can be used to alleviate the coincidence problem, as it was previously addressed. The proof that this cannot be done with non-interacting models can be found, e.g., in [43]. The most recent cosmological observations point to Ωm ' 0.3, Ωr ' 10−4 , ΩDE ' 0.7 and weff ' −0.7 today. The coincidence problem is attenuated if these values are postulated to correspond to a final state of an accelerating scaling solution, instead of a transition set of values. Taking the time-time-components of (2.55) in a FLRW universe (1.29), we get the usual continuity equations of the dark energy and matter species plus an interaction term which stands for the energy transfer between the interacting fluids: ρ˙ m + 3Hρm (1 + wm ) = −Q,

and ρ˙ DE + 3HρDE (1 + wDE ) = Q,

(2.56)

where, accordingly, Q stands for the time-component of the interaction vector, dots denote derivatives with respect to cosmic time t, H = a/a ˙ is the Hubble factor and w = p/ρ is the equation of state parameter. The sign of Q in (2.56) denotes the direction of the flow of energy transfer: if Q > 0 the matter fluid grants energy to DE whereas, if Q < 0, it is the DE fluid which sources the matter fluid. There are many models present in the literature which differ from the choice of the coupling term Q. The possibility of an interactive dark energy scalar field φ, coupled to a matter component and its respective cosmological consequences were seminally discussed in [174–177]. A related approach of interaction between dark matter and dark energy was proposed by Szydlowski [178], and a testing method was later developed in [179]. There is a plethora of interacting models of dark energy and matter, modelled as perfect fluids and studied in the context of dynamical systems [180–191]. When considering that the role of dark energy is played by a canonical scalar field we encounter the so-called coupled quintessence models which can also be studied in the context of dynamical systems and were first introduced and analysed in [166, 192, 193] as an extension of nonminimally coupled theories [193]. A variety of different models were soon proposed and are distinguishable by the form of the coupling [194–217]. The coupling function is usually taken to be a function of the Hubble parameter H, the scale factor a, the fluid and/or field’s energy density ρ, the field φ and/or its derivatives, etc. We are also interested in the case where the role of dark energy is played by a tachyon field. Coupled 33

tachyonic models, in which the dark energy component is allowed to interact with the matter sector have also been considered for different coupling functions [198, 218–221]. Most commonly, the coupling function is imposed at the level of the field equations and its cosmological implications are studied afterwards. However, the coupling could be naturally generated. In the previous section we have seen that a simple and well-motivated way to introduce a coupling between matter and the φ field, which stands for dark energy, is to perform a conformal/disformal transformation of the metric tensor. By doing so, the interaction is cast into a Lagrangian description. In particular, in the Einstein frame, we could have multiple components sourcing the right-hand side of the Einstein field equations, which, by conservation of the total energy momentum tensor (through conservation of the Einstein tensor, as described at the end of Section 1.3) could interact with one another. Generically, in the presence of two conformally/disformally related metrics, the action in the Einstein frame is cast as   ! Z q 2 √    X X Mpl (i) (j) S = d4 x −g R(gµν ) + Lφ (gµν , φ, ∇µ φ) + LM (gµν ) + −e g (j) Le(j) g e , ψ, ∇ ψ , µ m µν   2 i

j

(2.57) where i and j stand for the number of uncoupled and coupled fluids, respectively. φ stands for the dark energy field and ψ for the matter fluids present in the theory. Of course there is still some leeway when it comes to the choice of the conformal and/or disformal functions and the form of the potential V (φ). We will be mostly interested in the dynamical implications of such an interaction but a more detailed view on the phenomenological and observational properties can be found in [29, 172, 222–226]. Hereafter we will adopt g for the Einstein frame metric and ge and g¯ for a conformally and disformally related Jordan-frame metrics, respectively.

34

Chapter 3

Conformally Coupled Tachyonic Dark Energy Following the concepts introduced this far, in this Chapter we study a cosmological model where the role of dark energy is played by a tachyon scalar field. This field is coupled to the matter sector, at the level of the Lagrangian, by means of a conformal transformation of the metric tensor. The matter sector is assumed to be composed of a pressureless perfect fluid and an effective radiation-like perfect fluid, including both photons and relativistic neutrinos1 . Since we want to track the evolution of the three effective fluids - radiation, pressureless matter and dark energy - we make use of the tools of dynamical systems, in order to evolve the quantities with physical interest over time and extract the phenomenological information. The following scheme illustrates the methodology on which we rely in order to study the cosmological model, that is built in detail in the next section: • Find suitable variables and parameters to establish the dynamical system; • Identify the invariant sets of the system (including, if necessary, at infinity); • Perform a local analysis of the fixed points, including possible bifurcations; • Finally, restrict the parameter space with physical interest and extract information about the asymptotic past and/or future of the Universe.

3.1

The Model

Let us consider a four-dimensional spacetime manifold M endowed with a metric gµν . We are interested in studying how a conformal coupling between dark energy and matter can affect the dynamics 1 In fact neutrinos are now understood to be massive but, since this mass is constrained to be very small, they can effectively be treated as relativistic particles.

35

of the Universe. To do so, from now on we assume a conformal transformation of the metric from the Einstein frame to the Jordan frame gµν 7−→ geµν = C(φ)gµν ,

(3.1)

where C(φ) is the conformal coupling function and a tilde denotes quantities in the Jordan frame. From now on we will simply write C ≡ C(φ). As discussed in Section 2.2, we find g µν 7−→ geµν =

1 µν g , C

(3.2)

for the inverse of the conformal metric (3.1) and g ≡ det(gαβ ) 7−→ ge ≡ det(e gαβ ) = C 4 g,

(3.3)

for the determinant of the metric. We suppose that the field φ in the conformal transformation plays a role of dark energy and therefore the coupling between matter and dark energy is fully sketched if we consider that the gravitational Lagrangian depends on the metric gµν and the coupled matter Lagrangian depends on the conformal metric geµν defined in (3.1). We consider a scalar-tensor theory in the Einstein frame with action Z S=

 2  Z p √ MP g LeC (e gµν , ψ, ∂µ ψ), R + P (φ, X) + LM (gµν ) + d4 x −e d x −g 2 4

(3.4)

√ where MP ≡ 1/ 8πG is the reduced Planck mass and R is the Ricci scalar, which depicts the geometrical sector of the Universe and was defined in (1.23) in terms of gµν . The term Lφ ≡ P (φ, X) stands for the Lagrangian density of the scalar field (associated to dark energy). Conformally invariant fluids are included in LM . Finally, LeC is the Lagrangian for coupled fluids, allowed to depend on ψ and its derivatives, where ψ denotes matter fields propagating on geodesics defined by geµν . The relativistic fluids will be accounted for in LM and the effective matter fluid in LeC . As a first approximation, we assume the baryons to be uncoupled, and moreover, we ignore their contribution, due to to the stringent constraints from Solar System bounds [172]. This was properly addressed in Section 2.5 and is taken into account in most of the works present in the literature, since dark matter must stands for the majority of the pressureless matter sourcing the RHS of the Einstein equations. Throughout this chapter we will consider that the role of dark energy is played by a tachyon scalar field. This means that we will take the tachyonic Lagrangian defined in (1.51): P (φ, X) = Ltach ≡ −V (φ)

p √ 1 + ∂ µ φ∂µ φ = −V (φ) 1 − 2X, 36

(3.5)

where X = − 21 g µν ∂µ φ∂ν φ is the kinetic term associated to the tachyon scalar field φ, and V (φ) is a general self-coupling potential. It can immediately be noted that the assumption 1 − 2X ≥ 0 is needed in order to assure that the Lagrangian is real-valued. Variation of the action in (3.4) with respect to the metric gµν leads to the field equations in the Einstein frame   δS 1 2 φ M C = 0 =⇒ G ≡ R − g R = κ T + T + T µν µν µν µν µν µν , δg µν 2

(3.6)

where κ ≡ MP−1 is the scaled gravitational constant, Gµν is the Einstein tensor, Rµν is the Ricci curvature φ M and T C are the energy-momentum tensors for the scalar tensor, both computed from gµν , and Tµν , Tµν µν

field φ and the coupled and uncoupled fluids defined as p √ √ g LeC ) 2 δ( −gLM) 2 δ( −e 2 δ( −gLφ ) M C φ , Tµν ≡ − √ , Tµν ≡ − √ . Tµν ≡ − √ µν µν µν −g δg −g δg −g δg

(3.7)

From the expression of the tachyonic Lagrangian density, in (3.5), and the previous definition we gather that the energy momentum tensor for the tachyon field is φ Tµν = V (φ) √

p ∂µ φ∂ν φ 1 + ∂ α φ∂α φ. − V (φ)g µν 1 + ∂ α φ∂α φ

(3.8)

Let us recall that the Einstein tensor Gµν is still divergenceless, but now this does not imply the usual independent conservation relation for each energy momentum tensor on the RHS of (3.6). Actually, due to ∇µ Gµν = 0 and the conservation of the energy momentum tensor of the uncoupled fluids, it follows   φ C + Tµν = 0. ∇µ Tµν

(3.9)

As expected, since LeC depends on the field φ (through geµν ), the energy momentum tensors of dark energy and coupled matter are not separately conserved. The introduced coupling prevents the conservation of the individual energy-momentum tensors. The energy momentum tensor in the transformed frame is related to the one in the original frame defined in (3.7) as TCαβ where J ≡

p = √

gρσ eρσ −e g δe T = JC TeCαβ , −g δgαβ C

(3.10)

p √ −e g / −g is the Jacobian of the transformation, defined in terms of the determinants of the

metric, which are related by (3.3). To compute the equation of motion for the scalar field we take the usual procedure of varying the action with respect to φ (following the procedure described in Section 2.1) p √ Z Z g LeC ) δ( −gP ) 4 √ 4 δ( −e δS = δSφ + δSC = d x −g δφ + d x δφ = 0. δφ δφ 37

(3.11)

One can easily show that Z δSφ =

√

V d x −g p 1 + ∂ µ φ∂µ φ 4

  ∇µ ∂ ν φ∂µ φ∂ν φ V,φ φ − δφ, − 1 + ∂ µ φ∂µ φ V

(3.12)

and Z δSC =

p p Z  gµν −e g LeC ) δe −e g 4 d x δφ = d x C,φ gµν TeCµν δφ, δe gµν δφ 2 4

δ(

(3.13)

where the subscript (,) denotes partial derivatives. Combining equations (3.12) and (3.13), we obtain the coupled equation of motion p 1 + ∂ µ φ∂µ φ ∇µ ∂ ν φ∂µ φ∂ν φ V,φ φ − − = − Q 1 + ∂ µ φ∂µ φ V V

(3.14)

where  = ∇µ ∇µ is the D’Alembertian operator and  C 1  ,φ Q = J C,φ gµν TeCµν = TC , 2 2C

(3.15)

is the interaction term, where TC ≡ gαβ TCαβ is the trace of the energy momentum tensor. Even though the energy momentum tensor for each fluid is not independently conserved, from (3.8) and (3.14), we find: ∇µ Tφµν = −Q∂ ν φ. From (3.9), this translates into: ∇µ TCµν = Q∂ ν φ.

3.2

(3.16)

Background Cosmology

For cosmological applications we consider the homogeneous, isotropic, spatially flat FLRW metric defined in (1.29). For this model, we assume a perfect fluid (1.30) for both matter and the scalar field, which gives rise to energy-momentum tensors of the form: C Tµν = (ρC + pC )uµ uν + pC gµν ,

(3.17)

φ Tµν = (ρφ + pφ )uµ uν + pφ gµν ,

(3.18)

where ρC (ρφ ) and pC (pφ ) are, respectively, the energy density and pressure for the coupled fluid (field), in the Einstein frame and uµ is the fluid four-velocity for a comoving observer, which, by definition, satisfies uµ uµ = −1. Projecting equation (3.16) along the four-velocity uµ we find uµ ∇µ ρC + (ρC + pC )∇µ uµ = Quµ ∇µ φ. 38

(3.19)

We are interested in the Einstein frame dynamics of the time-dependent scalar field in the presence of radiation and matter. The dynamical equations arising from the Einstein field equations consist of two coupled differential equations for the scale factor a(t) and the functions ρ(t) and p(t). We can now write the coupled equation of motion, the fluid conservation equations and the Friedmann equations for our model in the Einstein frame:    3/2 V,φ 1  2 ˙ ¨ ˙ 3H φ + φ+ 1−φ = 1 − φ˙ 2 Q, V V

(3.20)

˙ ρ˙ C + 3HρC (1 + wC ) = −Qφ,

(3.21)

ρ˙r + 4Hρr = 0,

(3.22)

H2 =

κ2 (ρφ + ρC + ρr ) , 3

(3.23)

2

κ H˙ = − (ρφ (1 + wφ ) + ρC (1 + wC ) + ρr (1 + wr )) , 2 where an upper dot denotes time derivatives, H =

a˙ a

(3.24)

is the Hubble rate and the subscripts r and C stand

for radiation and coupled matter respectively. Again, wC = pC /ρC and wr = pr /ρr = 31 , are the EoS parameters for coupled matter and radiation, respectively. Note that we choose to represent radiation separately in equation (3.22) as it will always be conformally invariant. This issue was addressed in Section 2.2 and is ascribed to the fact that the trace of the energy momentum tensor for radiation is zero, implying that Q in equation (3.15) vanishes. The exchange of energy between the species becomes evident from (3.21) together with the corresponding continuity equation for the field: ˙ ρ˙ φ + 3Hρφ (1 + wφ ) = Qφ,

(3.25)

where wφ = pφ /ρφ is the EoS parameter for the tachyon field. From equations (3.21) and (3.25) we infer that, whenever Qφ˙ > 0, energy is being transferred from the matter sector to dark energy, whereas when Qφ˙ < 0, it is the dark energy fluid which concedes energy to the matter sector. Note that the assumption 1 − 2X ≥ 0 now translates into φ˙ 2 ≤ 1. Combining equations (3.8) and (3.18) leads to the following expressions for the energy density, pressure and EoS for the field: q V (φ) , pφ = −V (φ) 1 − φ˙ 2 and wφ = φ˙ 2 − 1. ρφ = q 1 − φ˙ 2

(3.26)

The tachyon field presents a very particular type of dynamics in a sense that, regardless of the steep39

ness of the potential, its equation of state parameter always varies between 0, in which case it behaves like a dust-like fluid, and −1, resembling a cosmological constant. Also, this model cannot feature fields with a phantom nature [29], since that would imply wφ < −1. From the analysis performed in (1.36) we directly conclude that the tachyon energy density evolves according to ρφ ∝ a−n with 0 ≤ n ≤ 3. In these conditions, we can also rewrite the interaction term as Q=−

3.3

C,φ (ρC − 3pC ) . 2C

(3.27)

Dynamical Equations

In order to study the evolution of the Universe under this model we reduce the above system of equations (3.20)-(3.24) and (3.27) to a set of first order autonomous differential equations. To do so, we introduce the following dimensionless2 variables [115]: 2 2 2 ˙ y 2 ≡ κ V , z 2 ≡ κ ρC , r2 ≡ κ ρr , α ≡ C,φ , λ ≡ − 1 V,φ , τ ≡ CC,φφ , x ≡ φ, 2 3H 2 3H 2 3H 2 CH κ V 3/2 C,φ (3.28)

The variable α stands for a normalization of the change in the conformal coupling in terms of the Hubble parameter. For what concerns this work we will consider λ and τ to be constants (note that we cannot simply take α to be a constant, since it also depends on the Hubble parameter which is dynamical). This choice corresponds to a scalar field potential and a conformal coupling function of the form:

V (φ) =

  V∗ ,

λ=0

  V02 ,

λ 6= 0

φ2

and

C(φ) =

  C1 eC2 φ , 1  C φ 1−τ , 0

τ =1 (3.29) τ 6= 1

where V∗ is a constant with units of (mass)4 , V0 ≡ 2/(κλ) and C2 are constants with units of mass, C1 1

is a dimensionless constant and C0 is a constant with units of (mass) 1−τ . By imposing τ to be a constant we are implicitly assuming a specific power-law dependence on the field for the conformal coupling function, if τ 6= 1, or an exponential form, if τ = 1. For tachyonic dark energy, the simplest closed system of autonomous equations is the one characterised by the inverse square potential (the constant potential with λ = 0 represents the trivial case, leading to a cosmological constant type of behaviour) and not by the exponential potential, in contrast with canonical quintessence [43]. This system has been previously studied for the particular case where α is identically zero, which corresponds to a totally uncoupled setting [115]. Making use of these variables, we can also define the density parameter and the equation of state parameter for each fluid: 2

For a dimensional analysis we refer the reader to Appendix A.

40

Ωφ = √

y2 , 1 − x2

(3.30)

ΩC = z 2 ,

(3.31)

Ωr = r 2 ,

(3.32)

wφ = x2 − 1.

(3.33)

The fact that the energy densities of matter and dark energy can be written in a clear way in terms of the dynamical variables (3.28) reinforces their physical meaning. Since we are concerned with cosmological applications, we wish to look for viable transitions between dark energy and dark matter dominated eras, respectively. For this reason, hereafter we will focus on the case of a dark energy fluid coupled to a dust-like (pressureless) perfect fluid for which, as described in detail in Section 2.2: wC = w eC = 0.

(3.34)

From (3.30)-(3.32), in order to preserve flatness in the Universe, we derive the Friedmann constraint, in terms of the dimensionless variables Ωφ + ΩC + Ωr = 1 =⇒ √

y2 + z 2 + r2 = 1, 1 − x2

(3.35)

which we used to replace z in terms of x and y, reducing the dimension of the dynamical system. From (3.27) and using the definition of the dynamical variables in (3.28), (3.34) and (3.35), the interaction term can be recast into   2κ2 y2 2 2 −r . Q = −α(1 − 3wC )z = −α 1 − √ 3H 3 1 − x2

(3.36)

Taking the definition of the interaction term (3.36) and the variables defined in (3.28), we can write the system of autonomous equations as   √ κ2 Q p 2 , x0 = (x2 − 1) 3x − 3λy − 1 − x 3H 3 y 2   y √ H0 y0 = − 3λxy + 2 , 2 H   H0 r0 = −r 2 + , H   H0 0 , α = α (τ − 1)αx − H

(3.37) (3.38) (3.39) (3.40)

where we have used H0 3 =− H 2



 p 1 2 2 2 1+ r −y 1−x , 3 41

(3.41)

Table 3.1: Couplings of a barotropic perfect fluid to a tachyonic dark energy component studied in the literature, with Q as defined in (2.56). Note that ρC and ρφ stand for the energy density of the barotropic perfect fluid and the dark energy field, respectively, wC is the EoS of the fluid and q stands for a constant. Q

Reference Gumjudpai et. al (2005) ([198]) Farajollahi & Salehi (2011) ([218]) Landim (2015) ([220]) Shahalam et. al (2017) ([221]) This work

˙

−qρC φ f,φ − f ρC (1 − 3wC )φ˙ − Hq ρC ρφ φ˙ −q ρ˙φ C,φ − 2C ρC (1 − 3wC )φ˙

where we replace the Einstein frame time coordinate, t, by the number of e-folds, N ≡ ln a, and a prime denotes derivatives with respect to N . For a single component flat Universe, from (1.33) and (1.34) it follows H0 H˙ 3 = 2 = − (1 + w). H H 2

(3.42)

Phenomenologically, we can define the effective equation of state parameter of the Universe, as if the Universe was composed of a single fluid: weff =

ptot = wC ΩC + wr Ωr + wφ Ωφ . ρtot

(3.43)

Following (3.42), the effective EoS parameter can be identified as the quantity inside the brackets in (3.41) such that, using the Friedmann constraint (3.35), p 3 1 H0 = − (1 + weff ) =⇒ weff = r2 − y 2 1 − x2 . H 2 3

(3.44)

Being a global parameter, it is the one from which we gather if the universe undergoes a period of accelerated (weff < −1/3) or decelerated (weff > −1/3) expansion. For this specific case we require − y2

p r2 1 1 − x2 + 0 in order to impose an expanding Universe. Given this and the fact that the dynamical system (3.37)-(3.40) has the invariant submanifolds y = 0, r = 0 and α = 0, we focus only on the non-negative values of y, r and α. Following the previous analysis, taking α ≥ 0 means that Q, given in (3.36), will always be non-positive. This means that whenever x < 0, which stands for φ˙ < 0, energy is being pumped from the matter sector into the dark energy component and when x > 0 the opposite holds (given Q 6= 0).

3.5

Dynamical System Analysis

Through the parametrisation described in (3.28) and (3.29), we obtain a 4D autonomous system of equations, (3.37)-(3.40), which can be studied in the context of dynamical systems. We focus only on the case where τ 6= 1 since, for τ = 1, the dynamical equation (3.40) decays very rapidly and there are no fixed points with α 6= 0. This means that the dynamical evolution is very close to the one in the uncoupled system [115] and moreover, the asymptotic behaviour is the same. For τ < 1 the conformal coupling function C(φ) takes the form of a power-law and, for τ > 1, of an inverse power law.

3.5.1

Fixed Points, Stability and Phenomenology

Recall that, for what concerns this work, we consider a coupled dust-like perfect fluid (baryonic matter and dark matter), i.e, a fluid characterised by pC = 0, in the presence of an ultrarelativistic fluid (photons and neutrinos) and a dark energy component. The fixed points for this system can be obtained by setting the left hand side (LHS) of the autonomous equations (3.37)-(3.40) equal to zero and are given in Table 3.2 where, for simplicity, we use √ yu ≡

λ4

+ 36 − 6

λ2



!1/2 and

√ xc ≡ 4 3 24 +

s

λ2 (3

− 4τ ) (τ − 1)2

2

−1/2 + 576

.

(3.52)

Note that the subscript (u) is used to label the variables for the fixed points (D) and (E) and the subscript (c) refers to the fixed point (F). In Table 3.2 we also present the corresponding values of the density parameter, (3.30), and equation of state parameter of the field, (3.33), for each fixed point. Additionally, in Table 3.3 we study the effective equation of state parameter together with the range of parameters corresponding to an accelerated expansion state, i.e., weff < −1/3, for the fixed points presented in Table 3.2. We are also interested in performing a stability study, and we do it through linear stability analysis, as described in Section 1.1.1, i.e., through the inspection of the eigenvalues (e1 , e2 , e3 , e4 ) of the 4 × 4 stability matrix M, defined in (1.4), and constructed from equations (3.37)-(3.40), by considering small 44

Table 3.2: Fixed points of the system (3.37)-(3.40) considering a dust-like coupled fluid, corresponding cosmological parameters and existence conditions, as defined in equations (3.30), (3.33), and (3.48). The form of yu and xc is given in (3.52). Note that, by construction, we have λ 6= 0 and τ 6= 1. Name

x

y

r

α

Ωφ

wφ

Existence

(O) (A± )

0 ±1

0 0

0 0

0 0

0 Und.

−1 0

α≡0 α≡0

(B± ) (C) (D)

±1 0

0 0 yu

1 1 0

0 0 0

0 0 1

0 −1 λ2 yu2 3 −1

∀λ, τ ∀λ, τ ∀λ, τ

(E)

λy √u 3

yu √

0

1

λ2 yu2 3

(3.53)

(F)

λy √u 3

xc

q

λ2 x2c +12

√

1−x2c −λxc

12(1−x2c )

0

√ 3λ 2(1−τ ) yu √ 2yc2 ( 3λyc −3xc )

√

1−x2c −yc2

2

√ yc

1−x2c

−1

x2c − 1

(3.56)

perturbations around each fixed point. Recall that first, if all the eigenvalues of the matrix M (evaluated at the fixed point) have positive real parts we speak of a repeller (or repelling node) as trajectories will be repelled from the fixed point. If all eigenvalues have negative real parts, the fixed point is said to be an attractor (or attracting node) as it will attract all nearby trajectories. Finally, if at least two eigenvalues have real parts with opposite signs, then the corresponding fixed point is unstable and is called a saddle point, which attracts trajectories in some directions but repels them along others. We present the stability and hyperbolicity character, as defined in Section 1.1.1, of each fixed point in Table 3.4. The stability study for the non-hyperbolic points is made in the next section through a bifurcation study, as it was delineated in Section 1.1.2. We also present the matrix elements of M in Appendix B and the corresponding eigenvalues for each fixed point in Appendix C. Note that we have a four-dimensional system and the stability analysis will be different from the three-dimensional uncoupled system. That is, even if the fixed point coordinates do not depend on the conformal coupling parameter α, its stability will be affected by the fact that the coupling is present in the system. Fixed points listed as (O)-(D) were also found in [114, 115] for the uncoupled system with α ≡ 0 and in [220] for a different coupling. For this reason, we chose to represent the new fixed points (E) and (F), for which α 6= 0, separately in Table 3.2. Note that the fixed points (O) and (A± ) are also presented separately in Table 3.2. This is ascribed to the fact that these three points are not defined for the coupled system, as they stand for singularities in the dynamical system (more specifically in equation (3.37))3 . Indeed, they are not formally part of the full phase space and their dynamical role can only be recovered when α ≡ 0. In that case, some comments can be stated: 3

One option would be to try to compactify the α = 0 plane.

45

Table 3.3: The effective equation of state parameter, weff as defined in (3.44), and the parameter region that leads to an accelerated expansion of the Universe, i.e., weff < −1/3, for the dust-like fluid fixed points of the system (3.37)-(3.40), as defined in Table 3.2. The existence conditions are also taken into consideration. The form of yu and xc is given in (3.52) and yc corresponds to the value of y for the fixed point (F).

weff Accelerated expansion

(O)

(A± )

(B± )

(C)

0

0

1 3

1 3

No

No

No

No

(D) λ2 yu2 3

−1

λ4 < 12

(E) λ2 yu2 3

−1

(3.53) ∧ λ4 < 12

(F) p −yc2 1 − x2c
(3.56) ∧ [ λ4 < 12 ∨ (1/2 < τ < 1)]

• The trivial point (O) represents a matter dominated solution, which is consistent with a null effective equation of state, weff = wC = 0. The scalar field is negligible near this fixed point. It will always be a saddle point, which attracts orbits along the x-axis and repels them towards the y-axis. • The dust-like points (A± ) can represent matter dominated solutions or scalar field dominated solutions, depending on the direction of the trajectories approaching them: for trajectories close to the boundary x2 + y 4 = 1 they represent scalar field dominated solutions, whereas for trajectories close to the boundary y = 0, they stand for matter dominated solutions. Accordingly, weff = wC = wφ = 0 and these fixed points are not capable of providing accelerated expansion. They are always saddle points. Actually, even though the points (A± ) and (O) are not formally present in the coupled system, numerically, we observe that, in their neighbourhood, the flow is still saddle-like, when compared to the uncoupled case (attracting trajectories in the x direction and repelling them in the y direction). Some comments can be stated regarding the existence and stability of the fixed points (B)-(F) found in Table 3.2 and are listed below. Note that since we wish to approach the problem from a cosmological point of view we need to check the existence of each fixed point according to the condition (3.48). • Points (B± ) stand for radiation-dominated solutions. They represent removable singularities in equation (3.37) which can be regularised and studied. They are characterised by Ωr = r2 = 1 and, consistently, a constant effective equation of state weff = wr = 1/3. Also, we find wφ = 0, meaning that near these points the scalar field acts as a dust-like fluid. Moreover, they are independent of the choice of the parameters and hence, can always exist, and we find that they are always repellers since all four eigenvalues are real-valued and positive. Being the only repellers, makes them the only possible past attractors of the system. • The radiation-dominated fixed point (C) is also a removable singularity of equation (3.37). It is characterised by weff = 1/3 and wφ = −1, which means that the scalar field (whose contribution, 46

in practical terms, is negligible) freezes and behaves as a cosmological constant. Its existence does not depend on the choice of parameters and so, it will always be present in the phase space. Fixed point (C) is always a saddle since all of the corresponding eigenvalues are real-valued and one of them, e1 , is negative and the other three are positive. • Point (D) is a scalar field dominated solution since Ωφ = 1 for every parameter value. This fixed point has an explicit dependence on λ, exists for every constant value of the parameters and it always lies on the boundary x2 + y 4 = 1. We find that wφ = weff =

λ2 yu2 3

− 1. Note that, for

λ = 0, i.e., for the case of a constant potential, the field behaves as a cosmological constant, with wφ = −1, near this fixed point. Whenever λ4 < 12, this fixed point always lies inside the area of the phase space where the universe undergoes accelerated expansion. Point (D) is the attractor for the totally uncoupled system, with α ≡ 0, but by turning on the coupling it acquires a saddle nature. • The new fixed point (E) corresponds to a scalar field dominated solution and it only differs from (D) in the value of the coupling α. This is ascribed to the form of the dynamical equation (3.37), which for x = xu and y = yu , vanishes regardless of the value of coupling. In that case, according to equation (3.40), there are two critical values of α, corresponding to two different fixed points. Since, for symmetry reasons, as discussed in Section 3.4, we are only considering the case where α ≥ 0, this fixed point is only defined in the parameter space for (λ > 0 ∧ τ < 1) ∨ (λ < 0 ∧ τ > 1) .

(3.53)

In these conditions we have Ωφ = 1, meaning that it always lies on the boundary x2 + y 4 = 1. In the same manner we find that wφ = weff and whenever λ4 < 12 this fixed point is inside the area of the phase space depicting accelerated expansion. Moreover, we will see that for a fixed value of λ, a transcritical bifurcation between (E) and (F) occurs, when τ is such that one of the eigenvalues, e4 in this case, passes through zero, changing (E)’s stability nature. This will translate into (E) being a stable attractor for (λ < 0) ∧ (τ > 1) ∨ (λ > 0) ∧ (τ ≤ H) ,

(3.54)

with H≡1−

λ4 +

Otherwise it is a saddle point. 47

p λ4 (λ4 + 36) . 72

(3.55)

Table 3.4: Dynamical stability for the fixed points of the system (3.37)-(3.40), as defined in Table 3.2, for a pressureless fluid. The stability analysis was made by taking into account the eigenvalues of the matrix M evaluated at each fixed point. The eigenvalues can be found in Appendix C. Point

Hyperbolicity

Stability

(O) (A± )

Hyperbolic Hyperbolic

Saddle Saddle

(B± ) (C) (D)

Hyperbolic Hyperbolic Hyperbolic

Repeller Saddle Saddle

(E) (F)

Depends on λ and τ Depends on λ and τ

Stable for (3.54) and saddle for (3.56) Stable for (3.56)

• Fixed point (F) corresponds to a conformal scaling fixed point as the cosmological parameters Ωφ , wφ and weff depend on the value of λ and τ . These type of solutions where the scalar field and matter energy densities scale with each other, that is, 0 < Ωφ < 1 and 0 < ΩC = 1 − Ωφ < 1, are termed scaling solutions. This means that the Universe evolves under the influence of both matter and the scalar field. Due to the restriction (3.48), this point is only present in the phase space when:

(λ > 0) ∧ (H ≤ τ < 1)

(3.56)

with H defined in (3.55). This fixed point can provide accelerated expansion whenever the conditions in Table 3.3 are met and it is stable for (3.56). This means that, when its cosmological existence is verified, (F) is the attractor of the system. Furthermore, (E) is a saddle point for (3.56) and an attractor otherwise. Thereby, for a specific value of λ, a transcritical bifurcation between (E) and (F) occurs, when τ is such that e4 passes through zero, changing (E) and (F)’s stability character.

By looking at the value of weff for fixed point (F) in Table 3.3, one interesting feature is that, for a fixed value of the parameters, we can now achieve an accelerating universe by means of this scaling solution. This is of great physical interest since it could, in principle, as it was addressed before (Section 2.5), relieve the cosmic coincidence problem, as an everlasting expanding solution with Ωφ ≈ 0.7 can now be achieved. We also note that (E) and (F) are not defined for τ → 1 as that implies α → +∞ in both cases. What we find is that there is indeed an additional fixed point in the limit x → 0, y → 1 and α → +∞, which in the limit τ → 1 coincides with (E) and (F). This will be addressed in greater detail further on through the compactification of the phase space. 48

3.5.2

Bifurcation

As it was previously addressed, the conformally coupled system features one bifurcation involving the fixed points (E) and (F)4 . Through inspection of the eigenvalues of the stability matrix, evaluated at the fixed point (E) (found in Appendix C), we infer that its stability character depends only on the value of e4 :   √ λ2 λ2 − λ4 + 36 (4τ − 5) e4 = − − 3. 24(τ − 1)

(3.57)

All of the other three eigenvalues are independent of τ and are negative ∀λ ∈ R. Hence, if e4 < 0, the fixed point (E) is an attractor, whereas if e4 > 0, it is a saddle. Accordingly, its stability character changes when: e4 = 0 =⇒ τ = 1 −

λ4 +

p λ4 (λ4 + 36) , 72

(3.58)

which, according to the local analysis performed in the previous section for each fixed point, coincides with the value of τ for which the fixed point (F) enters the phase space, H, as defined in (3.55). This, together with a proper analysis of the numerical behaviour of the system, allows us to recognise that the fixed points (E) and (F) undergo a transcritical bifurcation when τ takes the value in equation (3.58) (for each specific value of λ), which corresponds to the bifurcation point. Even though we have not presented the eigenvalues for the fixed point (F), we conjecture that one of its eigenvalues switches sign at the bifurcation point. This type of behaviour is typical of a transcritical bifurcation and was verified numerically. The bifurcation diagram is illustrated in Figure 3.1 (a) for the particular case of λ = 2.3.

3.5.3

Physical Phase Diagram

As it was previously addressed, the system is not defined for τ = 1. This translates into the asymptotic limit α → +∞, as τ → 1, for points (E) and (F) (see Table 3.2). We find that there is indeed an additional fixed point with (x, y, α) → (0, 1, → +∞) which, in the limit τ → 1, coincides with (F), giving rise to the only attractor in the system. This leads to the need of compactifying the phase space. As we have already seen, the four-dimensional dynamical system (3.37)-(3.40) is invariant under the transformation (x, y, α) 7→ (−x, −y, −α) and moreover, the (x, y, α)-phase space is non-compact because: − 1 ≤ x ≤ 1, 0 ≤ y ≤

p 4

1 − x2 , 0 ≤ α < +∞.

In order to compactify the phase space we introduce the variable 4

For more details regarding the theory of dynamical bifurcations we refer to Section 1.1.2.

49

(3.59)

α*

F

4

E

λ

2

0

-2

E

-4 -4

τ (a) Transcritical bifurcation diagram

-2

0 τ

2

4

(b) Attractor according to the parameter region

Figure 3.1: Panel (a): Bifurcation diagram for the transcritical bifurcation, as described in Section 3.5.2, between the fixed points (E) (blue) and (F) (red) for λ = 2.3. A solid (dashed) curve represents a family of stable (unstable) fixed points. The vertical lines represent the flow generated by equation (3.40). Panel (b): Illustration of the attractor in each parameter region for different values of λ and τ , as described in Section 3.5.3. In the grey region, given in (3.54), the attractor of the system is the fixed point (E), whereas in the black region, given in (3.56), it is the fixed point (F). The white region stands for the parameter space where the dynamics evolves towards the attractor at the infinity of the coupling α. A ≡ arctan α,

(3.60)

as it was previously done in [63]. The phase space becomes compact with − 1 ≤ x ≤ 1, 0 ≤ y ≤

p 4

1 − x2 , 0 ≤ A < π/2,

(3.61)

making it possible to draw the global portrait of the phase space, including the asymptotic behaviour as α → +∞. Actually, we find that, in the parameter region where neither (E) nor (F) can exist in the phase space, i.e., whenever (λ > 0 ∧ τ > 1) ∨ (λ < 0 ∧ τ < 1), there are attracting fixed points at the infinity √ of the coupling. When this is the case, we find that the entire boundary y 2 = 1 − x2 is composed of fixed points and, depending on the value of the parameters, one of them becomes the global attractor of the system. We gather that, for (λ > 0 ∧ τ > 1), the attractor is x → 0, y → 1, r → 0 and α → +∞ whereas for the parameter space (λ < 0 ∧ τ < 1) the attractor is x → xu , y → yu , r → 0 and α → +∞. In Figure 3.1 (b), we present a depiction of the attractor of the system according to the parameter region. Note that, for each possible value of the parameters there is always at most one finite attractor of the system. In this sense, in order to avoid the presence of asymptotic diverging behaviour in the model, we will only focus on the following parameter space: (λ > 0 ∧ τ < 1) ∨ (λ < 0 ∧ τ > 1) . 50

(3.62)

Regardless of the value of the parameters and which fixed point is the attractor, in the parameter space defined in (3.62), there are always, at least, 5 different fixed points present in the phase space: (C) and (D), which are saddle points, (B± ) which are repellers and possible past attractors and (E), an attractor or a saddle point. In Figure 3.2 we present the phase portrait of the uncoupled system and the two-dimensional reduced phase portraits for the coupled system (for λ = 3 and different values of τ ). The latter are constructed from the coupled dynamical system given by equation (3.37)-(3.40) by assuming that the value of α is constant and equal to its value at the attractor of the system in each case (red point). Even though this synthetic representation does not reproduce the real system (since the value of α is also dynamical), it is a simple way of drawing a qualitative analysis of the dynamics of the system to compare with the uncoupled case. Regarding the analysis done before, we summarise the dynamical behaviour in each region of the physical parameter space (3.62): • For (λ < 0) ∧ (τ > 1) or (λ > 0) ∧ (τ < H), with H defined in (3.55), there are 5 fixed points present in the phase space (grey region in Figure 3.1 (b)). Qualitatively, by turning on the coupling (α 6= 0), (D) turns into a saddle and a new attractor (E) arises in the phase space, which differs only on the value of the coupling. For this reason it gives rise to the same asymptotic behaviour, therefore, giving rise to the same physical properties. Note that the attractor is always located in the (x, y) boundary of the phase space. The fixed point (E) is only found inside the region which generates accelerated expansion for λ4 < 12. There is only one attractor and two possible past attractors. Asymptotically, when τ → ±∞ and α → 0, the dynamics corresponding to the uncoupled system is recovered, which is to say that (D) and (E) coincide and become the only attractor of the system. In Figure 3.2 (b), a qualitatively depiction of the dynamics in this parameter region is presented for λ = 3 and τ = −10. • For (λ > 0) ∧ (H ≤ τ < 1), with H given in (3.55) which coincides with the existence region for the fixed point (F), there are 6 fixed points present in the phase space (black region in Figure 3.1 (b)). Qualitatively, (E) turns into a saddle and a new attractor (F) emerges into the phase space, with the same value of the coupling. Depending on the value of τ , the point (F) can be located anywhere starting from the fixed point (E), along an arc of the parabola (x, y, α) = (xc , yc , αc ), towards the point (0, 1, αc ), as τ −→ 1. The fixed point (F) is only found inside the yellow re

gion, denoting accelerated expansion of the Universe, for [ λ4 < 12 ∨ λ4 > 12 ∧ 1/2 < τ < 1 ]. Even if the attractor itself does not meet the previous condition, there are many solutions which feature a transient period of accelerated expansion by crossing the yellow region. In the limit case where τ = H, the fixed points (E) and (F) coincide and become the attractor of the system. On the other hand, when τ → 1 (from below), we recover the attractor at the infinity of α. In Figures 3.2 (c) and (d), a qualitatively depiction of the dynamics in this parameter region is presented for 51

y

1.0

1.0

0.8

0.8

0.6

D

y

0.4 0.2

0.6

D/E

0.4 0.2

0.0 A/B -1.0

O/C

-

-0.5

0.0 A/B -1.0

A/B+

0.0 x

0.5

(a) α ≡ 0

y

O/C

-

1.0

-0.5

A/B+

0.0 x

0.5

1.0

(b) α ≡ 0.13, τ = −10

1.0

1.0

0.8

0.8

0.6

F

D/E

y

0.4 0.2

F

0.6

D/E

0.4 0.2

0.0 A/B -1.0

O/C

-

-0.5

0.0 A/B -1.0

A/B+

0.0 x

0.5

O/C

-

1.0

(c) α ≡ 1.5, τ = 0

-0.5

0.0 x

A/B+

0.5

1.0

(d) α ≡ 6.1654, τ = 0.7

Figure 3.2: Panel (a): Phase portrait of the dynamical system given by equations (3.37) and (3.38) for the purely uncoupled system (α ≡ 0) with λ = 3. Panels (b)-(d): Two-dimensional reduced phase portraits of the coupled dynamical system given by equation (3.37)-(3.40) (see text), for λ = 3 and different values of τ . The black and red points correspond to the fixed points defined in Table 3.2. Note that, for these values of α, the effects of the points (O) and (C) are not visible. The grey and yellow regions correspond to the existence region, (3.48), and the region where the Universe undergoes accelerated expansion, (3.45). λ = 3 and τ = 0 and τ = 0.7, respectively. Taking the previous analysis into consideration, the dynamics with physical interest occurs whenever (λ > 0) ∧ (H ≤ τ < 1) and the scaling solution (F) is allowed to exist and be the attractor of the system.

3.6

Viable Cosmologies

Finally, we seek viable cosmologies, i.e., trajectories on the phase space capable of reproducing the expansion history of the Universe. To do so, we take into account the parameter space with physical meaning, described in the latter section. Considering all the fixed points presented in Table 3.2 (including the fixed points for the purely uncoupled system), the transition radiation −→ matter −→ dark energy could be cosmologically viable considering: (B± ) or (C) −→ (A± ) or (O) −→ (D), (E) or (F), 52

35 30

α=0 τ=0.95

25 20 α

15 10 5 0 -10

-5 ln a

0

5

Figure 3.3: Left Panel: Phase portrait of the dynamical system given by equations (3.37)-(3.40) for λ = 3 and τ = 0.95. The variable “A” is defined in (3.60). The solid line corresponds to the trajectory of the coupled system and the dashed line to the uncoupled system. Right Panel: Illustration of the evolution of the value of α according to the fine tuning of initial conditions, as described throughout the text: in the past α is approximately zero and the dynamics in the past resembles the dynamics of the uncoupled system. At a given time instant, the coupling α starts to grow leading to the conformally coupled scenario. Initial conditions: xi = 2 × 10−6 , yi = 6 × 10−9 , ri = 0.998463, αi = 8 × 10−10 . respectively. However, (C) is always a saddle point, meaning that it cannot be the past attractor. Thus, the evolution of the Universe at early times could only be given by points (B± ), the only repellers in the system, characterised by weff = 1/3. Also, the fixed points (A± ) and (O) can only exist when α is identically zero in the dynamical system. According to the fact that α = 0 is an invariant set of the system, the only way to have α → 0 at a future attractor is to take α = 0 from the beginning5 . This means that the dynamics corresponding to the fixed points of the uncoupled system, (O)-(D), can only be recovered in these conditions. The only fixed points depicting radiation and matter dominance in the Universe are fixed points with α = 0. This implies that, in this framework, radiation and matter domination can only be achieved in the past if α → 0 and y → 0 at early times, corresponding to the direction of matter domination given by the fixed points (A± ). However, these fixed points are not formally present in the phase space for the coupled system. Through numerical simulation, we find that in the neighbourhood of (A± ), the flow is still saddle-like (as it was the case in the uncoupled system). When this is the case, the dynamics will be affected by the fixed points of the uncoupled system and, at a given instant, the coupling α is able to grow leading to the conformally coupled scenario with (E) or (F) as possible future attractors (see Figure 3.3). However, the dynamics corresponding to the uncoupled case (see Figure 3.2 (a)) is never fully recovered since, for the coupled system, the line y = 0 represents a singularity (see Figure 3.2 (b)). Alternatively, it is also possible to take large values of the product τ x in (3.40) (since H 0 /H ≤ 0), leading to a strictly decreasing value of the coupling, until it eventually reaches zero. But, since −1 ≤ x ≤ 1, the system would have to be characterised by a large value of the parameter τ , in which case, either the value of α decreases towards zero, and we recover α → 0 in the future, or it diverges towards the attractors at the infinity (depending also on the sign of λ). 5

53

0.4

1.0

0.2

0.8

0.0

0.6 Ω

Ωϕ (α=0) Ωc (α=0)

w

Ωr (α=0)

0.4

Ωϕ (τ=0.9)

weff (α=0)

-0.4

wϕ (α=0) weff (τ=0.9)

-0.6

Ωc (τ=0.9)

0.2

-0.2

Ωr (τ=0.9)

wϕ (τ=0.9)

-0.8

0.0

-1.0

-10

-5 ln a

0

5

-10

-5 ln a

0

5

Figure 3.4: Example of the evolution of the relative energy densities and the EoS parameters according to the conformally coupled tachyonic model with τ = 0.9 and λ = 3. The solid curves stand for the dynamical evolution according to the uncoupled model and the dashed ones for the coupled one, for the same set of initial conditions: xi = 2 × 10−6 , yi = 2 × 10−11 , ri = 0.998463, αi = 1 × 10−10 . Additionally, depending on the value of λ, the fixed point (F) is only defined in a very short window of values of τ , according to (3.56). This suggests that this conformally coupled tachyonic model can only be cosmologically viable for a specific set of initial conditions, leading to the problem of fine tuning of initial conditions. For the admissible parameter space, described in the previous section, and the fine tuning of the initial conditions, we find that the late time attractor is always (E) or (F) and the past attractor is always (B+ ). Furthermore, initially, equation (3.37) is always negative and thus, the dynamics close to the x-axis is monotonically decreasing (in fact it decreases abruptly, evolving very rapidly towards (B− )). Hence, all the trajectories in the physically viable phase space correspond to heteroclinic orbits (defined in Section 1.1) which start from (B+ ) and end in (E)/(F). In Figure 3.4 we present one example of the evolution of the energy densities, as defined in (3.30)(3.32), and the corresponding evolution of the effective EoS parameter and the EoS parameter for the field, as defined in (3.44) and in (3.33), respectively. The solid lines correspond to the uncoupled system (α ≡ 0 and λ = 3) and the dashed lines correspond to the coupled system (τ = 0.9 and λ = 3).

3.7

Effective Potential

An immediate consequence of the energy exchange between the fluids is that an interaction term will be present in the expression for the energy density of the pressureless coupled fluid. This can be derived by integration of the continuity equation for the matter fluid (3.21), with the coupling Q given by (3.27), leading to:  ρC = ρC,0

a a0

−3 

φ φ0



1 2(1−τ )

,

(3.63)

where ρC,0 and φ0 are constants, a0 ≡ 1 is the value of the scale factor today, and the field φ can 54

be expressed in terms of the potential defined in (3.29). This can be studied, according to the present observational constraints, in order to restrict the possible values of the parameters λ and τ . It is also possible to define an effective potential associated with the scalar field by rewriting the coupled equation of motion (3.20) as: 

φ¨ + 1 − φ˙ 2



3H φ˙ +

V,φeff

!

V eff

= 0,

(3.64)

where, for a pressureless matter sector, V,φeff V eff

q V,φ = + V

1 − φ˙ 2 C,φ V

2C

ρC ,

(3.65)

with ρC given in (3.63). By doing so, equation (3.64) resembles the uncoupled equation of motion. Taking the inverse square potential and a power-law conformal coupling function, as described in (3.29) (with τ 6= 1), the collective effect of both the potential associated to the scalar field and the coupling is described by an effective potential of the form: q    ˙2 ρ 2+n 1 − φ φ φ C,0  V eff (φ) = V0eff exp −2 ln + φ0 5 − 4τ V02 a3 φn0 

where n ≡

1 2(1−τ )

and V0eff =

V02 φ20

(3.66)

is a constant of integration with units of (mass)4 . With this definition,

the first term in the exponential in (3.66) corresponds to the standard inverse square potential whereas the second term accounts for the effect of the coupling. This means that, when coupled to the matter sector, the evolution of the φ field is driven by the properties of the effective potential, which becomes “matterdependent”. Accordingly, when τ 6= 1, the direction of energy exchange depends upon the evolution of the field and the sign of τ . For simplicity, we can also write (3.66) as q  ˙2 ρ 2+n 1 − φ C,0 φ  = V (φ) · Z(φ), V eff (φ) = V (φ) exp  5 − 4τ V02 a3 φn0

(3.67)

where V (φ) is the inverse square potential defined in (3.29) and we have defined q  1 − φ˙ 2 ρC,0 φ2+n . Z(φ) ≡ exp  5 − 4τ V02 a3 φn0

(3.68)

From (3.67) it is clear that, taking τ → ±∞, which corresponds to the limit in which the coupling vanishes, i.e., α → 0, we recover the inverse square potential V (φ). In Figure 3.5 (left panel) we present an illustration of the composition of the effective potential, constructed from the combination of the inverse square potential and the coupling. 55

Veff

� (ϕ)

1

� (ϕ) ����(ϕ)

0.100

τ = 0.6 τ = 0.5 τ = 0.4

0.010 0.001 ϕ

0

20

40

60

80

ϕ

Figure 3.5: Left Panel: Illustration of the composition of the effective potential from the combined effect of the inverse square potential and the coupling, as described in (3.67). This qualitative representation is made in logarithmic scales for the specific case where n = 1/4. Right Panel: Illustration of the effective potential as described in (3.66) for different values of τ and a constant value of B = 105 as defined in (3.70). Decreasing the value of τ has the effect of shifting the value of the minimum to greater values of φ (in absolute value). The minimum point for each curve is marked with a black dot.

Note that we will implicitly assume that

q

1 − φ˙ 2 ∼ O(1) near the fixed point, as a first approxima-

tion. The value of φ˙ is, according to the analysis performed in the previous sections, related to the value of the parameters λ and τ . Motivated by the study of the parameter space with physical interest performed in the previous section, we take τ < 1 and ρ ≥ 0. In this case, it is easy to verify that V eff is a function with one minimum value corresponding to 2−2τ

φm ' [B (1 − τ )] 5−4τ

(3.69)

with B ≡ 4V02 a3

φn0 , ρC,0

(3.70)

and φ ∈ R \ {0}. The position of the minimum depends on the scale factor, a, and on the parameters λ, through V0 , and τ . All the other variables are considered fixed. An illustration of the shape of the effective potential for different values of τ and B = 105 is presented in Figure 3.5 (right panel). It is easy to see that, by decreasing the value of τ the minimum of the potential is shifted to larger values of φ (in absolute value). When τ → −∞, the coupling vanishes and the minimum of the potential is shifted to φ → +∞, which is consistent with a potential with no minimum value, as was the case of the uncoupled system. This is a reflection of the fact that the scaling solution is only allowed to exist when the effective potential has global minimum values. Also, consistently, the minimum value of the effective potential, presented in (3.69), is only valid when τ < 1, 56

corresponding to a power-law form for the conformal coupling function (see (3.29)). When τ > 1 the conformal coupling function takes the form of an inverse power law, which has no effective minimum.

3.8

Summary

We can conclude that this conformally coupled model, where dark energy is allowed to interact with the matter sector, allows for two possible attractors, as described in Table 3.2: point (E) or (F), depending on the (λ, τ ) parameter region. We saw that point (E) differs from point (D) only on the value of the coupling, therefore giving rise to the same qualitative behaviour that was found for the uncoupled system. Both points (D) and (E) represent scalar field dominated solutions, meaning that the Universe will unavoidably evolve to a state of total scalar field dominance. Hence, the novelty associated to this model lies in the existence of point (F), corresponding to a scaling solution. Since current observations suggest that we live in an accelerated expanding Universe with Ωφ ' 0.7 and weff ' −0.7 then, all three points could, in principle, represent solutions capable of describing the present observed Universe. Furthermore, the scaling solution (F) could be used to alleviate the cosmic coincidence problem, allowing for the description of an everlasting Universe with Ωφ ' 0.7 near the fixed point. Also we found that the equation of state parameter for the tachyon field always varies between 0 and −1 and, for point (F), its value is a function of the parameters implying that, in principle, it could be fitted in order to match observational constraints. Also, for every parameter space there is only one possible attractor of the system which always stands for scalar field dominated/scaling solutions. However, we find that the cosmological history of the Universe can only be reproduced in specific conditions, since past radiation and matter domination eras can only be achieved if α → 0 and y → 0 at early times. This suggests that the problem of fine tuning of initial conditions is still present in this model. Consequently, in these conditions, a full description of the expansion history of the Universe is possible, in the parameter region where all the trajectories are attracted towards point (F) and it features an accelerated expanding-type of behaviour. In order to further address the question of viable cosmologies we need to dive into perturbation theory. This is work to be carried out in the future in order to restrict this model according to the observational constraints.

57

58

Chapter 4

Disformally Coupled Quintessence Following the treatment in the previous chapter, the main goal of this ongoing work is to perform a detailed analysis of generalised couplings (as general as we find possible) between a canonical scalar field, portraying dark energy, and the matter sector. To do so, we assume that the scalar field is disformally coupled to a perfect fluid, as was done in [63], but we extend the analysis in the existing literature by assuming that the disformal coefficient can also depend on the kinetic term associated to the scalar field. This has already been discussed in [227] but we intend to revisit it with a different approach. We find novel solutions when considering this new kinetic term dependence, expressed through an associated parameter, µ. As expected, when µ = 0 we recover the previously studied setting. We obtain a more general set of solutions in the sense that µ acts as a parameter which can be used to increase the region of parameter space that render a fixed point stable.

4.1

The Model

Consider a four-dimensional spacetime manifold M endowed with a metric gµν . Now we are interested in studying the cosmological consequences of introducing a coupling between dark energy and matter, emerging from a disformal rescaling of the metric tensor. From now on, we will assume a disformal transformation of the metric from the Einstein frame to the Jordan frame (see Section 2.4): gµν 7−→ g¯µν = C(φ)gµν + D(φ, X)∂µ φ∂ν φ ,

(4.1)

in which case, we can also define the inverse transformed metric g µν 7−→ g¯µν =

D∂ µ φ∂ ν φ 1 µν g − 2 C C − 2CDX

(4.2)

and the determinant of the metric g 7−→ g¯ = C 3 (C − 2DX) g 59

(4.3)

where C(φ) and D(φ, X) are the conformal and disformal coupling functions respectively and X = − 12 ∂µ φ∂ µ φ is the kinetic energy associated with the scalar field. The cosmological consequences of the interaction between dark energy and matter due to a disformal transformation have been vastly studied for the case where both the conformal and disformal coefficients depend only on the field φ but generically, the functions C and D can also depend on the kinetic term X. It has been shown that if C also depends on X, Ostrogradski’s instabilities arise in some theories [121, 124]. Hereafter we take the field φ in the disformal transformation to play the role of dark energy and, similarly, the coupling between matter and dark energy is fully described if we consider that the gravity Langrangian depends on the metric gµν and the coupled matter Lagrangian depends on the disformal metric g¯µν defined in (4.1). The action in the Einstein frame becomes Z S=

 Z √ MP2 d x −g g L¯D (¯ gµν , ψ, ∂µ ψ), R + P (φ, X) + LM (gµν ) + d4 x −¯ 2 4

√



(4.4)

√ where MP ≡ 1/ 8πG = 2.4×1018 GeV is the reduced Planck mass, R is the Ricci scalar which depicts the geometrical sector of the Universe and is defined in terms of gµν , Lφ ≡ P (φ, X) is the Lagrangian of the scalar field (associated to dark energy), LM is the Lagrangian for uncoupled matter and L¯D is the Lagrangian for coupled matter, allowed to depend on ψ and its derivatives, where ψ denotes matter fields propagating on geodesics defined by g¯µν . Similarly to the previous case, we will consider only couplings to dark matter and radiation and, as a first approximation, discard the contribution of the baryonic sector, due to the the stringent constraints coming from Solar System bounds [172]. Throughout this work we will consider P (φ, X) ≡ X − V (φ) where V (φ) is a general self-coupling potential associated to the scalar field φ. Variation of the action in (4.4) with respect to the metric gµν leads to the field equations in the Einstein frame   1 δS φ M D 2 = 0 =⇒ G ≡ R − g R = κ T + T + T µν µν µν µν µν µν , δg µν 2

(4.5)

where κ2 ≡ MP−2 ≡ 8πG, Gµν is the Einstein tensor, Rµν is the Ricci curvature tensor, both computed φ M and T D are the energy-momentum tensors for the scalar field φ and matter from gµν and Tµν , Tµν µν

defined as φ Tµν

√ √ √ 2 δ( −gLφ ) 2 δ( −gLM) 2 δ( −¯ g L¯D ) M D √ √ ≡ −√ , T ≡ − , T ≡ − . µν µν −g δg µν −g δg µν −g δg µν

(4.6)

The energy momentum for the canonical scalar field is φ Tµν

 = ∂µ φ∂ν φ − gµν

 1 α β gαβ ∂ φ∂ φ + V (φ) . 2

60

(4.7)

The Einstein tensor Gµν is divergenceless, but in our theory this does not imply the usual independent conservation relation for each energy momentum tensor on the RHS of (4.5). Actually, due to ∇µ Gµν = 0 and the conservation of the energy momentum tensor of ordinary matter, we have   φ D ∇µ Tµν + Tµν = 0.

(4.8)

The energy momentum tensors of dark energy and coupled matter are not individually conserved. This is ascribed to the fact that L¯D also depends on the field φ (through g¯µν ). The energy momentum tensor in the transformed frame is related to the one in the original frame defined in (4.6) as   √ gρσ ¯ρσ −¯ g δ¯ 1 αβ α β α β TD = √ TD = Cδρ δσ − D,X ∂ φ∂ φ∂ρ φ∂σ φ J T¯Dρσ , −g δgαβ 2 where J ≡

√

√ −¯ g / −g and the subscript (,) denotes partial derivatives and, accordingly √ g L¯D ) 2 δ( −¯ D ¯ . Tµν ≡ − √ −¯ g δ¯ g µν

(4.9)

(4.10)

To compute the equation of motion for the scalar field we take the usual procedure of varying the action with respect to φ Z δS = δSφ + δSD =

√ δP δφ + d x −g δφ 4

Z

√ δ( −¯ g L¯D ) d x δφ = 0. δφ 4

One can easily show that (see Appendix D): Z √ δSφ = d4 x −g(φ − V,φ )δφ. And also (see Appendix D): √ √ Z Z gµν g L¯D ) δ¯ −¯ g ¯µν 4 δ( −¯ 4 δSD = d x δφ = d x TD (C,φ gµν + D,φ ∂µ φ∂ν φ) δgµν δφ 2 √  √   √ √ −¯ g ¯µν −g −¯ g ¯µν ω − −g∇µ √ TD D∂ν φ + ∇ω ∂ φ √ TD D,X ∂µ φ∂ν φ δφ. −g 2 −g

(4.11)

(4.12)

(4.13)

Combining equations (4.11), (4.12) and (4.13), we obtain the coupled Klein-Gordon equation φ = V,φ − Q,

(4.14)

where  = ∇µ ∇µ is the D’Alembertian operator and
1
1 Q = J T¯Dµν (C,φ gµν + D,φ ∂µ φ∂ν φ) − ∇µ J T¯Dµν D∂ν φ + ∇ω J∂ ω φT¯Dµν D,X ∂µ φ∂ν φ , (4.15) 2 2 is the interaction term, unveiling how energy flows between the dark energy component and the matter sector. If we have a transfer of energy between the fluids, from (4.7) and (4.14), we find: ∇µ Tφµν = 61

−Q∂ ν φ. In order to preserve general covariance, equation (4.8), the coupled matter fluids have to counterbalance this exchange: ∇µ TDµν = Q∂ ν φ.

(4.16)

For the trace of the energy momentum tensor we have, from (2.47) and using (4.1): TD = gαβ TDαβ = JCgρσ T¯Dρσ + JD,X X∂ρ φ∂σ φT¯Dρσ .

(4.17)

It is also useful to define: Tk ≡ ∂α φ∂β φTDαβ = J(C − 2D,X X 2 )∂ρ φ∂σ φT¯Dρσ .

(4.18)

From (4.17) and (4.18) we find the following relation for the energy momentum tensor in the transformed frame: TDαβ D,X ∂ α φ∂ β φ αβ ¯ + Tk . TD = CJ 2CJ(C − 2D,X X 2 )

(4.19)

Using the relations (4.18) and (4.19) it is possible to write (4.15) in the unbarred frame.

4.2

Background Cosmology

In the same manner as before, we perform calculations in the Einstein frame and, to do so, we consider a homogeneous, isotropic, spatially flat FLRW metric in Cartesian coordinates, as described in (1.29). We also assume that the various matter-energy components of the Universe are well described, at large scales and with high precision, by a continuous perfect fluid. Accordingly, the energy-momentum tensors, for the coupled fluids and the field, are considered to take the form: D Tµν = (ρD + pD )uµ uν + pD gµν ,

(4.20)

φ Tµν = (ρφ + pD )uµ uν + pφ gµν ,

(4.21)

where ρD (ρφ ) and pD (pφ ) are the fluid (field) energy density and pressure respectively and uµ is the fluid four-velocity, which for a comoving observer is given by uµ = (−1, 0, 0, 0). Taking this and projecting equation (4.16) along the four-velocity uµ we get uµ ∇µ ρD + (ρD + pD )∇µ uµ = Quµ ∇µ φ. 62

(4.22)

The scalar field is assumed to depend only on time and coordinate time derivatives will, consistently, be denoted by an upper dot. The Einstein field equations give rise to two coupled differential equations for the scale factor a(t) and the functions ρ(t) and p(t). Taking (4.14), (4.22) and (4.5), we are in a position to write the modified Klein-Gordon equation, the fluid conservation equation and the Friedmann equations for our model in this frame: φ¨ + 3H φ˙ + V,φ = Q,

(4.23)

˙ ρ˙D + 3HρD (1 + wD ) = −Qφ,

(4.24)

H2 =

k2 (ρφ + ρD ), 3

(4.25)

2

k H˙ = − (ρφ (1 + wφ ) + ρD (1 + wD )), 2 where again H =

a˙ a

(4.26)

is the Hubble rate and wD = pD /ρD is the equation of state parameter for the

coupled fluid. Note that, as an initial step, for what concerns this work, we will assume that there is only one effective fluid coupled to the dark energy field. In future work we intend to consider the effect of two coupled fluids (radiation fluid and pressureless fluid). Again, the exchange of energy becomes more apparent when writing the continuity equation for the field: ˙ ρ˙φ + 3Hρφ (1 + wφ ) = Qφ,

(4.27)

where wφ = pφ /ρφ is the EoS for the field. Equations (3.25) and (3.21) define the direction in which energy is being transferred: if Qφ˙ > 0, it is the matter sector which grants energy to DE whereas if Qφ˙ < 0 it is the dark energy fluid which sources the matter sector. Comparing (4.21) and (4.7) it is possible to infer the expression for the energy density and pressure of the field 1 1 ρφ = φ˙ 2 + V (φ) and pφ = φ˙ 2 − V (φ), 2 2

(4.28)

and the EoS for the field becomes wφ =

1 ˙2 φ + V (φ) pφ = 21 . ˙2 ρφ 2 φ − V (φ)

(4.29)

In the limit where the potential energy dominates, we recover the behaviour of a cosmological constant, with wφ = −1, and, on the other hand, when the evolution is governed by the kinetic energy, the EoS becomes wφ = 1. This means that the value of wφ varies as a function of the kinetic energy, 21 φ˙ 2 , and the potential energy, V (φ): wφ ∈ [−1, 1]. 63

Taking equations (4.23) and (4.24) it is possible to rewrite the interaction term in this frame (Appendix E).

4.3

Dynamical Equations

In order to study the evolution of the Universe under this model we reduce the above system of equations (4.23)-(4.26) to a set of first order autonomous differential equations. To construct the dynamical equations it is useful to introduce the following dimensionless1 variables (some of them seminally introduced in [59]): κ2 φ02 κ2 V κ2 ρD DH 2 1 V,φ 2 , y2 ≡ , z ≡ , σ ≡ , λV ≡ − , 2 2 2 6 3H 3H Cκ κ V D,X X D,XX X 2 1 D,φ 1 D,φX X λD ≡ − , µD ≡ , ηD ≡ , ξD ≡ − , κ D D D κ D x2 ≡

λC ≡ −

1 C,φ , κ C (4.30)

where we replace the Einstein frame time coordinate t by the number of e-folds, N ≡ ln a, and derivatives with respect to N are denoted by a prime. For concreteness, we assume that the conformal and disformal coupling coefficients and the scalar field potential have the following forms: C(φ) = C0 e2ακφ ,

D(φ, X) =

e2(α+β)κφ µ X , M 4+4µ

V (φ) = V0 e−λκφ ,

(4.31)

where α, β, λ and µ are all considered to be dimensionless constant parameters and M , V0 and C0 are constant parameters with mass, mass to the fourth, and no dimensions respectively. This choice corresponds to the simplest case, where the quantities λV to ξD , in (4.30), are constants. Note that the novelty associated with this model lies in the possibility of having the disformal function depending on the kinetic term of the scalar field. We restrict the model to the case of a power-law dependence, expressed through the new parameter µ and moreover, we will consider only positive values of µ in order to avoid possible singularities in the coupling (when X passes through zero). We are interested in understanding how the presence of this parameter affects the dynamics of the system. Recall that we had already stated that the exponential potential is the simplest case for quintessence. With this choice, by comparing (4.30) and (4.31), it is easy to conclude that the freedom associated to the system lies in 4 parameters only: λV = λ, λC = −2α, λD = −2(α + β), µD = µ, ηD = µ(µ − 1), ξD = −2µ(α + β). (4.32) Taking the variables defined in (4.30), along with the choice for the form of the coupling functions and the potential in (4.31), we can write the system of dynamical equations, which is ensured to be 1

For a dimensional analysis we refer the reader to Appendix A.

64

autonomous and closed2 : x

0

y0 z0 σ0

r     3 κQ H0 2 x+ λV y + , =− 3+ H 2 3H 2 ! r r 3 2 H0 = λV x + y, 2 3H ! r 2 H 0 1 3 κQ x 3 1 + wD + z, =− + 2 3H 3 2 H 2 z2   √ x0 H0 = 6(λC − λD )x + 2µD + 2 (1 + µD ) σ, x H

(4.33) (4.34) (4.35) (4.36)

where H0 3 = − (2x2 + (1 + wD )z 2 ), H 2

(4.37)

subjected to the Friedmann constraint x2 + y 2 + z 2 = 1,

(4.38)

which we use to remove z from the dynamical equations (4.33), (4.34) and (4.36), reducing the number of autonomous equations of the system. The quantity in brackets in equation (4.37) defines the effective equation of state parameter such that: H0 3 = − (1 + weff ) =⇒ weff = x2 − y 2 + wD (1 − x2 − y 2 ), H 2

(4.39)

Recall that the effective equation of state is the parameter from which we gather if the Universe undergoes a period of accelerated (weff < −1/3) or decelerated (weff > −1/3) expansion. In this case, we require: x2 − y 2 + wD (1 − x2 − y 2 ) < −1/3,

(4.40)

which depends solely on the dynamical variables x and y and on the EoS of the coupled fluid. Time integration of (4.37) gives the evolution of the scale factor over time, at any fixed point (xf , yf , σf ) of the phase space: 2

a ∝ t 3(1+weff ) ,

(4.41)

which corresponds to a power-law solution, with weff as defined in (4.39). Note that this means that even if we are not able to describe the entire evolution of the Universe, the asymptotic behaviour will always be well-defined, provided that it is given by a specific fixed point solution. 2 Indeed, if we choose a different functional form for either one of them, we require a set of evolution equations for the functions λV , λC , λD , µD , ηD and ξD . This issue is discussed in [63].

65

Using the expression for Q given in Appendix E (for a general form of the potential and the coupling functions), the definition of the dynamical variables in (4.30) and (4.31), the interaction term can be rewritten, as a function of the parameters, α, β, λ and µ, as: n h√
κQ −1 2 6x 1 + 3µ + 2µ2 + wD (µ + 1) − x2 (−4α = F −α (1 − 3w ) z + 3σ D 2 3H h√
 6x3 5µ − µ2 +6αµ (1 − 2wD ) + 2 (α + β) (1 + 2µ)) − y 2 λ 1 + 3µ + 2µ2 z 2 + 18σ 2
 +wD µ (µ − 1)) − x4 2µ (α + β) − 6αµ2 wD − x2 y 2 λµ (µ + 1) z 2 , (4.42) with 
   F ≡ 1 + 3σ z 2 1 + 3µ + 2µ2 − 2x2 (1 + 3µ) + 18σ 2 z 2 µ (1 + µ) + 2x2 µ (1 + 2µ) x2 . (4.43) As expected, by setting µ = 0 in (4.42) we recover the coupling function introduced in [63], for a purely field-dependent disformal transformation. One immediate conclusion is that the coupling function given in (4.42) has a second order dependency on the variable related to the disformal coupling, σ, whereas in [63] only a linear dependence was found. This will, in principle, introduce new (disformal) fixed points. Other useful quantities are Ωφ = x2 + y 2 ,

(4.44)

x2 − y 2 , x2 + y 2

(4.45)

1 − 6σx2 , 1 − 6µσx2

(4.46)

wφ =

wD = w ¯D

where w ¯D is the equation of state parameter of the fluid in the frame defined by the metric g¯µν (where the particles travel along geodesics). Note that for a disformal transformation of the metric the equation of state parameter becomes frame-dependent, except for the exceptional case of the system with µ = 1. For example, in the case of dust and radiation, w ¯D = 0 and w ¯D = 1/3 respectively. It follows that for dust-like (pressureless) fluids, the equation of state parameter vanishes in both frames. For radiation-like fluids, the EoS parameter in the Einstein frame becomes: wD =

1 1 − 6σx2 . 3 1 − 6µσx2

We further introduce the parameter Z, related to the Jacobian J as r r −¯ g D 2 J= =C 1 + g µν ∂µ φ∂ν φ = C 2 Z 2 , −g C which, in this frame, reads 66

(4.47)

(4.48)

Z=

p 1 − 6σx2 .

(4.49)

From (4.48), we require Z 6= 0 in order to avoid metric singularities. Recall that, for simplicity, we will consider a single coupled fluid with equation of state parameter wD . We also perform a parametrisation of the EoS parameter to γ≡w ¯D + 1,

(4.50)

such that 0 ≤ γ ≤ 2. From (4.38), the meaning of the dynamical variables (4.30) x and y is straightforward: x2 plays the role of the relative kinetic energy density of the field, while y 2 stands for the potential energy density. Together they compose the total energy density parameter of the scalar field, given by (4.44).

4.4

Phase Space and Invariant Sets

The physical phase space, i.e., the invariant set which only contains orbits with physical meaning, can be restricted by considering that the energy density of the matter fluids is always positive, implying that 0 ≤ Ωφ ≤ 1, which translates into: 0 ≤ x2 + y 2 ≤ 1.

(4.51)

This condition defines an unitary circle on the (x, y)-plane, centred at the origin. Inspecting the dynamical system of equations (4.33), (4.34) and (4.36), we find that y = 0 is an invariant set of the system which, furthermore, is invariant under the transformation y 7−→ −y.

(4.52)

This implies that the resulting dynamics for negative values of y is an exact copy of the positive y case. For this reason, throughout this work, we focus only on positive values of y. Taking this into account, the physical phase space in the (x, y)-plane reduces to a half-unit disk centred at the origin. According to (4.44), points on the unit circle stand for scalar field dominated universes (i.e., points for which Ωφ = 1). Furthermore, the system is invariant under the simultaneous transformation (x, λ, α, β) 7−→ (−x, −λ, −α, −β).

(4.53)

In other words, the phase space is fully described if we take into account only positive values of λ, for instance. Note that the presence of the coupling does not allow for more symmetries. The (x, y, σ)-phase space is still non-compact because: 67

− 1 ≤ x ≤ 1, 0 ≤ y ≤

p 1 − x2 , 0 ≤ σ < +∞.

(4.54)

This calls for the need to compactify the phase space and, to do so, similarly, we introduce the variable Σ ≡ arctan σ.

(4.55)

The phase space becomes compact in the region − 1 ≤ x ≤ 1, 0 ≤ y ≤

p 1 − x2 , 0 ≤ Σ < π/2,

(4.56)

which represents a semi-circular prism of length π/2. This parametrization will be useful whenever we wish to draw the global phase space, including the asymptotic behaviour as σ → ∞.

4.5

Dynamical System Analysis

The fixed points with an arbitrary constant equation of state parameter, γ, can be found by setting the LHS of the autonomous equations (4.33), (4.34) and (4.36) equal to zero and solving the resulting polynomial equations for x, y and σ. Furthermore, according to the definition of the variables in (4.30) and the choice for the form of the potential and the coupling functions in (4.31), this system can easily be reduced to previously studied cases, on which we will base our analysis: • If µ = 0 the system reduces to disformally coupled quintessence, in the case where the conformal and disformal functions can only depend on the field itself. This was studied in [63]. • If µ = 0 and β → −∞ this system coincides with the conformally coupled quintessence case. This was properly studied in [43, 192, 198]. • Finally, if µ = 0, β → −∞ and α = 0 we are left with the standard uncoupled quintessence scenario, presented in [43, 59]. In Appendix F we present the fixed points for the system where the disformal function only depends on the field, corresponding to setting µ = 0 in equations (4.33), (4.34) and (4.36), as described above. The fixed points listed correspond to a fluid with an arbitrary constant equation of state parameter γ. We do so in order to make the comparison between the two systems easier.

4.5.1

Fixed points, Stability and Phenomenology for a pressureless fluid

First, we take into account the possibility of having a non-relativistic fluid disformally coupled to the dark energy fluid. Accordingly, we take γ = 1, which implies a pressureless fluid: wD = 0 ⇔ pD = 0. 68

Table 4.1: Fixed points of the system (4.33)-(4.36) for the case γ = 1 (labelled d) and corresponding cosmological parameters as defined in equations (4.44), (4.45) and (4.49). In fixed points (E± )d and (F± )d , the expressions σE/F± represent the solutions of the polynomial for σ given in the main text. Name

x

y

σ

Ωφ

wφ

Z

(A± )d

±1 q − 23 α

0

0

1

1

1

0

2 2 3α

1

1

−1

1

(B)d .

√λ q6

(C)d

3 2

(D)d √

(E± )d (F± )d

√

α+λ

2β 2 −3(1+µ)2 3(1+µ) √ √ 2β+ 2β 2 −3(1+µ)2 √ 3(1+µ) 2β−

√

0 q 1− q

λ2

0

1

0

α2 +αλ+3 (α+λ)2

α(α+λ) − 3+α 2 +αλ

1

0

σE±

x2E

1

0

σF±

x2F

1

p 1 − 6σE± Ωφ p 1 − 6σF± Ωφ

6

3 +α2 +αλ 2

√

λ2

(α+λ)2

3

The fixed points of the system are obtained by setting the LHS of (4.33), (4.34) and (4.36) equal to zero. They are labelled (A)d − (F)d and are registered in Table 4.1. For a complete analysis we also list the relevant cosmological parameters, Ωφ , wφ and Z. Note that we will use a subscript d when referring to the fixed points obtained by consideration of a dust-like fluid. The pairs σE± and σF± represent the solutions of a second order polynomial for the corresponding value of xE and xF , respectively, listed in Table 4.1: i h √ 
54µ (3µ + 1) x5 + 36 6µx4 (α + β) − 54µ (µ + 1) x3 σ 2 + −9x3 2µ2 + 9µ + 3 i √ √ −6 6x2 (αµ + α − 2βµ − β) − 9 (µ + 1) (2µ + 1) x σ + 6α + 3x = 0

(4.57)

This choice of representation for the fixed points is ascribed to the fact that the analytical solutions obtained for σ, when solving the second order equation (4.57) are complicated and reveal little information. In Table 4.2 we study the effective equation of state parameter together with the range of parameters which render accelerated expansion for the dust-like fixed points, i.e., which satisfy weff = x2 − y 2 < −1/3. This means that the fixed points that correspond to accelerated expansion of the Universe can only exist when y 6= 0, leading to wφ 6= 1. Without any further information, by inspection of Table 4.1, this restriction immediately tells us that the only fixed points capable of providing an accelerated expanding description are (C)d and (D)d . We are also interested in the study of the stability character of each fixed point and, similarly to what was done in the previous Chapter, we do it through analysis of the eigenvalues (e1 , e2 , e3 ) of the matrix M, constructed by consideration of small perturbations around each fixed point. The eigenvalues obtained for each fixed point can be found in Table G.1 in Appendix G and here we only treat fixed points (A)d -(D)d as the remaining have complicated dependences on the parameters and a more careful study has to be made. In general, the stability character of each fixed point has a clear dependence on 69

Table 4.2: The effective equation of state parameter, weff as defined in (4.39), and the parameter values that lead to an accelerated expansion of the Universe, i.e., weff < −1/3, for the dust-like (γ = 1) fixed points of the system (4.33)-(4.36). The conditions listed throughout the text for the existence of the fixed points are also taken into consideration.

weff Accelerated expansion

(A± )d

(B)d

1 No

2 2 3α

No

(C)d λ2 3 λ2

−1 λ/2

x2E No

(F± )d x2F No

the parameter range in which the study is being carried out. Throughout this analysis we rely on the fact that 0 ≤ Ωφ ≤ 1, which translates into a non-negative energy density for the fluid: ρφ ≥ 0. Finally, taking into account the possible values the parameters can take, some comments can be stated regarding the existence and stability of the fixed points (A)d − (F)d , found in Table 4.1. • Points (A± )d are independent from the introduced parameters and so, they can always exist. Since x is the only non-zero dynamical variable, we speak of scalar field kinetic dominated solutions. Based on equation (4.45), they are also called kination fixed points, as they represent dominance of kinetic energy over potential energy. They are characterised by a stiff equation of state for the field, wφ = 1 and, as expected, as Z = 1 these fixed points present no metric singularity and can be treated under this formalism. They also feature a constant effective equation of state, weff = 1, and so are not capable of describing an accelerating behaviour. The stability of (A± )d depends on the value of the parameters α, β, λ and µ, being free to be an attractor, a repeller or a saddle point. (A+ )d is an attractor for r α 6 ∧ β< 2

r

√ 3 ∧ λ< 6 ∧ β> 2

r

3 (1 + µ), 2

(4.58)

3 (1 + µ). 2

(4.59)

and a repeller for r α>−

Correspondingly, (A− )d is an attractor for r α>

r √ 3 3 ∧ λ− (1 + µ), 2 2

and a repeller for 70

(4.60)

r α
− 6 ∧ β 4. It is characterised by a constant equation of state parameter for the field, wφ = 1/3 and, being conformal, presents no metric singularity. The effective equation of state is, accordingly, weff = 1/3, meaning that this fixed point cannot provide accelerating behaviour. It is found to be either a stable spiral, an attractor or a saddle point as the existence condition λ ∈ R \ [−2, 2] renders negative real parts for e1 and e2 . We find that it is an attractor for     8 8 ∧ β λ ∨ 2 < λ ≤ √ 15 15 75

(4.69)

• Points (D± )r and (E± )r are disformal fixed points. We find that points (D± )r and (E± )r can only √ √ exist for β ≥ 6 and β ≤ − 6, respectively. They are scaling fixed points in the sense that the density parameter of the field is a function of the parameters (β). The effective equation of state is also a function of the parameters but we find that these fixed points cannot portrait accelerated expanding solutions. Moreover, the EoS of the field is constant, wφ = 1. Through numerical simulation, we conjecture that (D− )r and (E− )r are always saddle points and moreover, in the range β>

√

6∧λ>β

(4.70)

(D+ )r is an attracting solution and when √ β < − 6 ∧ λ < β.

(4.71)

(E+ )r is also an attractor. For the case of a disformally coupled radiation fluid with µ = 1, we still recover totally disjoint attracting regions for each fixed point. This means that, for each value of the parameters there is one, and only one, possible attractor of the system.

4.6

Conclusions

We are only interested in viable cosmological trajectories, i.e., trajectories on the phase space capable of describing the expansion history of the Universe: start in a radiation dominated era, then unfold to a matter dominated era, and finally reproduce the present accelerated expanding Universe, by means of the dark energy component. Recall that this is an ongoing work and, as a first approximation, we have only considered the case of a single fluid coupled to the dark energy component: a pressureless effective fluid or a relativistic fluid. This allows us to have a qualitatively grasp of the dynamics for the two-fluid case, as in the past history of the Universe, the contribution of non-relativistic matter is negligible whereas in the present, it is the contribution of the relativistic fluids which is practically insignificant. Thus, in the asymptotic past, we should look for the fixed points corresponding to a Universe totally dominated by radiation. At late times, however, we wish to look for a combination of the effect of the dark energy and pressureless fluids. Furthermore, the scalar field should only play a relevant role at late times. This means that, at early times, and in order to avoid signatures of early dark energy, the variables associated to the scalar field (x and y) should be close to zero. Note that the complete full two-fluid dynamical analysis would 76

depend on 8 parameters. For simplicity of the analysis, in this primary approach, we have focused only on the case of a linear dependence on the kinetic term, i.e., µ = 1, in the disformal coupling function for the radiation fluid, which translates into an invariant equation of state parameter under the disformal transformation (see equation (4.47)). Regarding the previous analysis, the possible past attractors are (A± )r and the only possible late-time fixed points, that allow for accelerated solutions, are the scalar field dominated fixed point (C)d and the conformal scaling fixed point (D)d . The main different resides in the nature of the asymptotic behaviour for the solutions (C)d and (D)d . (C)d stands for a Universe completely dominated by the dark energy fluid at the fixed point. When (C)d is the attractor of the system, the coincidence problem, associated to the current observed values of the energy densities, persists. On the other hand, when the attractor is the scaling fixed point (D)d , the energy densities of the interacting species scale with each other and remain constant (for a fixed value of α and λ), independently of the choice for the initial conditions. This could, in principle, alleviate the coincidence problem. However, here we are only focusing on the asymptotic behaviour and so, we lack information about the possible trajectories throughout the history of the Universe. This will be better addressed in the two-fluid study. In order to restrict the physical phase space, it would also be important to study the compactified system and discard possible parameters which lead to asymptotic behaviour towards possible attractor at the infinity of σ. Note that the disformal fixed points are never capable of describing an accelerated expanding Universe. They can only have a transient effect of delaying the trajectories towards the scalar field dominated or the conformal scaling attractor solution. Moreover it could lead to momentarily periods of energy transfer between the coupled fluids. Therefore, in principle, there will be no new candidates for late time attractor, in comparison with the conformally coupled quintessence scenario [43, 192, 198]. This result was also found for the disformally coupled scenario in which the disformal coupling is only allowed to depend on the field [63]. This analysis simply shows that the introduction of the dependence of the disformal function in the kinetic term only seems to introduce new disformal fixed points, which are not capable of describing accelerated expanding solutions and therefore can not be the late time attractor for viable late time cosmologies. However, through the complete two-fluid study, new transient fixed points could be found, leading to different trajectories in the phase space. Furthermore, it was found that the kinetic disformal parameter µ can be used to widen the region of parameters that render stability to a given fixed point of the system.

77

78

Chapter 5

Final Remarks In this dissertation, we have introduced and discussed the construction of two different cosmological models, where the role of dark energy is played by a scalar field. Moreover, the DE φ-field is allowed to couple to the matter sector by means of a conformal/disformal transformation of the metric tensor. For each model we have adopted a dynamical system methodology. More specifically, we have rewritten the main cosmological equations as a dynamical system, constructed from a set of useful (with physical meaning) dimensionless variables. Furthermore, we have identified the invariant sets, including, if necessary, at infinity, and performed a local analysis of the fixed points, including possible bifurcations. Finally, by restricting the parameter space with physical interest, we were able to extract information about the past/future evolution of the Universe. First, we have studied a model where dark energy is represented by a tachyon scalar field, conformally coupled to a non-relativistic fluid (dark matter). The introduction of the coupling in the system, allows for two new possible attractors, which are related by means of a bifurcation in the parameter space. We further saw that, for every parameter region, there is only one possible attractor of the system, which stands for a scalar field dominated/scaling solution. In particular, the scaling solution can be used to alleviate the cosmic coincidence problem. However, we have found that this model is still not cured from the issue of fine tuning of initial conditions, since past radiation and matter dominated eras can only be achieved for a specific set of conditions at early times. Next, we have focused on a model where DE is driven by a canonical scalar field, coupled to the matter sector through a disformal transformation of the metric tensor. We have extended the analysis in the existing literature by introducing a dependence on the kinetic term of the scalar field in the disformal factor. Since for disformally coupled models, relativistic fluids are also allowed to couple to dark energy, we have started by studying couplings to a single fluid. This allows us to have a first grasp over the past and future asymptotic behaviours associated to this setting. We have assumed a power-law dependence on the kinetic term, expressed through a new parameter, µ. We collect that new disformal solutions arise 79

and, moreover, µ can be used to shape the parameter region of stability for each fixed point. Depending on the value of the parameters, any of the critical points can have a relevant cosmological role throughout the history of the Universe. Furthermore, we find that there are only two possible late-time attractors capable of reproducing the accelerated expansion of the Universe: a scalar field dominated fixed point and a scaling solution, which again, could be used to alleviate the cosmic coincidence problem. Because this is an ongoing work, we have primarily focused on the construction of the dynamical system and on acquiring intuition over the cosmological consequences. In order to describe possible physical trajectories, we will further compactify the phase space, look for global attractors of the system and study the complete two-fluid system. We intend to do this, first for the case with µ = 0, and then for µ 6= 0 and see how the introduction of the kinetic dependence on the disformal coefficient affects the overall dynamics of the system. In the future we plan to complete the analysis by studying the evolution of the perturbations associated to both models, in order to investigate the impact of the current observational constraints on the physical observables.

80

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96

Appendix A

Natural Units In order to make the calculations cleaner, it is universal practice in Cosmology to adopt the system of natural units where c = ~ = 0 = µ0 = 1.

(A.1)

By doing so, the meaning of the equations becomes more straightforward and there is a more intuitive grasp on the order of magnitude of cosmological quantities. By taking natural units we are collectively fixing the units of mass, length, time and charge relative to each other. This leaves only one unit free to fix them all which, by fundamental reasons, is usually taken to correspond to the dimensions in which energy is expressed, GeV, or equivalently: [E] = M,

(A.2)

where the square brackets denote the dimensions of the quantity inside them and M stands for dimensions of mass. This dimensional relation follows immediately from the relation between energy and mass, e.g. E = mc2 , with c = 1. Also, this implies that time and length should have the same dimensions, which from ~ = 1 reads: [T ] = M −1

(A.3)

[L] = M −1 .

(A.4)

and

For the energy density, defined as energy per unit volume, we find [ρ] = M 4 . 97

(A.5)

Following this, the dimensions of other useful cosmological variables can be derived   a˙ = M, [H] = a [κ] =

h√

i 8πG = M −1 ,

(A.6)

(A.7)

where H is the Hubble factor and κ is the scaled gravitational constant. Since the action Z S=

d4 xL

(A.8)

is a dimensionless quantity, we find for the Lagrangian density [L] = M 4 .

(A.9)

The dimensions of a scalar field can be deduced by quantum field theory arguments. For quantum canonical fields the dimensions follow from the free-field Lagrangian density: 1 1 Lfree ≡ ∂µ Φ∂ µ Φ − m2 Φ2 , 2 2

(A.10)

[Φ] = M.

(A.11)

leading to

In fact, because of the kinetic term given in terms of partial derivatives of the field, all bosonic fields in 4-dimensional spacetimes have canonical dimensions of mass. Another example are the fermionic fields like the Dirac field ψ(x) with free Lagrangian density given by ¯ (iγ µ ∂µ − m) Ψ. Lfree ≡ Ψ

(A.12)

This means that for the Dirac fields   ¯ = M 3/2 . [Ψ] = Ψ

(A.13)

In fact, because of the presence of the kinetic term with two fields and one partial derivative, all types of fermionic fields in 4-dimensional spacetimes have canonical dimension mass3/2 . For non-canonical scalar fields the dimensions could become non-standard. For instance, the Lagrangian density for the tachyon field is defined as 98

Ltach ≡ −V (φ)

p 1 + g µν ∂µ φ∂ν φ.

(A.14)

Since V stands for the potential energy density associated to the field, [V ] = M 4 ,

(A.15)

from where we conclude that the dimensions for the tachyon field are [φ] = M −1 .

99

(A.16)

Appendix B

Conformally Coupled Tachyonic Dark Energy - Linear Stability Matrix Consider the system with a tachyon scalar field coupled to a non-relativistic barotropic fluid (see Chapter 3. The stability analysis in Table 3.4, for the fixed points found in Table 3.2, is made through consideration of the eigenvalues of the 4 × 4 matrix M, evaluated at each fixed point (the eigenvalues can be found in Appendix C). For the dust-like fluid, with x0 , y 0 , r0 and α0 as defined in equations (3.37)-(3.40), the matrix elements Mij are: √
√1−x2
0 2 = 3αx 1 − r 3λy −3 − x α − 9x + 2 M11 = ∂x 2 ∂x 2y √ √ 2 −1 α r 2 −1 2 − 3λy 3 0 x 1−x ( )( ( ) ) M12 = ∂x ∂y = y3 M13 =

∂x0 ∂r

M14 =

∂x0 ∂α ∂y 0 ∂x ∂y 0 ∂y ∂y 0 ∂r ∂y 0 ∂α

M21 = M22 = M23 = M24 =

3/2

= = = =

αr(1−x2 ) y2 √ (x2 −1)((r2 −1) 1−x2 +y2 ) − 2  √ 2y 1 2 −2y 3λ − √3xy 2 1−x  √ √ 1 2 2 y 2 − 2 3λxy r − 9 1 − x 2

=0 3rxy 2 √ 2 1−x2

M31 = M32 =

∂r0 ∂y

M33 =

∂r0 ∂r

√ = −3r 1 − x2 y   √ = 21 3r2 − 3 1 − x2 y 2 − 1

M34 =

∂r0 ∂α

=0

M41 =

∂α0

M42 =

∂α0 ∂y

M43 =

∂α0 ∂r

M44 =

∂α0 ∂α

∂x



= ry

∂r0 ∂x

+3

=

= 12 α



2 √3xy 1−x2

 + 2α(τ − 1)

√ = −3α 1 − x2 y

= αr   √ = 12 r2 − 3 1 − x2 y 2 + 4α(τ − 1)x + 3 .

100

Appendix C

Conformally Coupled Tachyonic Dark Energy - Eigenvalues of the Stability Matrix Here we present the eigenvalues of the matrix M, as defined in Appendix B, evaluated at each fixed point, as presented in Table 3.2 (see chapter 3). The eigenvalues are used to infer the stability character of each fixed point, as described in 1.1.1. The eigenvalues for the fixed point (F) are not presented due to their extensive expression. For the fixed points which are only formally defined for the three-dimensional uncoupled system, i.e., when α ≡ 0, we find the following eigenvalues: • Fixed point (O) e1 = −3, e2 = 23 , e3 = − 21 • Fixed points (A± ) e1 = 6, e2 = 32 , e3 = − 21 For the fixed points of the four-dimensional coupled system, defined for α 6= 0, we list the eigenvalues below: • Fixed points (B± ) e1 = 6, e2 = 2, e3 = 2, e4 = 1 • Fixed point (C) e1 = −3, e2 = 2, e3 = 2, e4 = 1 101

• Fixed point (D) 2  √ 1 e1 = − 12 λ2 − λ4 + 36   √ 1 2 e2 = − 12 λ λ2 − λ4 + 36   2  √ 1 12 + λ2 − λ4 + 36 e3 = − 24   2  √ 1 36 + λ2 − λ4 + 36 e4 = − 24 • Fixed point (E)     √ 1 e1 = − 12 λ2 λ2 − λ4 + 36 + 24   √ 2 e2 = λ12 λ2 − λ4 + 36     √ 1 e3 = − 12 λ2 λ2 − λ4 + 36 + 36 e4 = −

√ λ2 (λ2 − λ4 +36)(4τ −5) 24(τ −1)

−3

102

Appendix D

Disformally Coupled Quintessence Equation of Motion In chapter 4, we have seen that the equation of motion for the field, φ, is computed through variation of the total action (4.4), according to φ: Z δS = δSφ + δSD =

√ δP d x −g δφ + δφ 4

Z

√ g L¯D ) δ( −¯ d x δφ = 0. δφ 4

(D.1)

Let us first consider the compuation of δSφ :   Z √ δP ∂P ∂P 4 √ δφ = d x −g δφ + δ (∇ω φ) = δSφ = d x −g δφ ∂φ ∂ (∇ω φ)   Z
1 ργ 4 √ ω ω d x −g −V,φ δφ − g ∂γ φδρ + ∂ρ φδγ δ (∇ω φ) = 2   Z
1 ρ ω 4 √ γ ω ∂ φδρ + ∂ φδγ δ (∇ω φ) = d x −g −V,φ δφ − 2   Z 1 ω 4 √ ω d x −g −V,φ δφ − (∂ φ + ∂ φ) δ (∇ω φ) . 2 Z

4

(D.2)

Considering δ (∇ω φ) = ∇ω (δφ) and according to the rule of integration by parts it is possible to write: Z δSφ =

√ d4 x −g [−V,φ δφ − ∇ω (∂ ω φδφ) + ∇ω (∂ ω φ) δφ] .

(D.3)

Taking into account Stokes’ Theorem, and assuming that the fields vanish at the boundary (infinity), the terms corresponding to total divergences also vanish, leading to: Z δSφ =

4

√

ω

d x −g [−V,φ + ∇ω (∂ φ)] δφ =

Z

√ d4 x −g [−V,φ + φ] δφ.

where  = ∇ω ∇ω is the D’Alembertian operator. An analogous treatement is done for the computation of δSD : 103

(D.4)

√


√ Z −¯ g L¯D δ¯ gµν gµν −¯ g ¯µν δ¯ 4 δSD = d x δφ = d x TD δφ = δ¯ gµν δφ 2 δφ   √ Z gµν ∂¯ gµν −¯ g ¯µν ∂¯ d4 x TD δφ + δ (∇ω φ) = 2 ∂φ ∂ (∇ω φ)   √ Z gµν ∂¯ gµν ∂X −¯ g ¯µν ∂¯ 4 TD d x δφ + δ (∇ω φ) = 2 ∂φ ∂X ∂ (∇ω φ) √ Z −¯ g ¯µν TD [(C,φ gµν + D,φ ∂µ φ∂ν φ) δφ d4 x 2   
1 γρ ω ω δ (∇ω φ) = + (D,X ∂µ φ∂ν φ − 2Dgµν ) − g ∂γ φδρ + ∂ρ φδγ 2   √ Z
1 −¯ g ¯µν d4 x TD (C,φ gµν + D,φ ∂µ φ∂ν φ) δφ + − D,X ∂µφ ∂ν φ ∂ ρ φδρω + ∂ γ φδγω 2 2  

1 + D δµγ δνρ + δνγ δµρ ∂γ φδρω + ∂ρ φδγω δ (∇ω φ) = 2 √ √ Z −¯ g ¯µν −¯ g ¯µν 4 d x TD (C,φ gµν + D,φ ∂µ φ∂ν φ) δφ − TD D,X ∂µ φ∂ν φ∂ ω φδ (∇ω φ) 2 2  √
−¯ g ¯µν ω ω TD D ∂µφ δν + ∂ν φδµ δ (∇ω φ) . + 2 Z

4

δ

(D.5)

Accordingly, considering δ (∇ω φ) = ∇ω (δφ) and applying the rule of integration by parts: √ √  Z −¯ g ¯µν −¯ g ¯µν 4 ω δSD = d x TD (C,φ gµν + D,φ ∂µ φ∂ν φ) δφ − ∇ω TD D,X ∂µ φ∂ν φ∂ φδφ 2 2 √   √   √ −¯ g ¯µν −¯ g ¯µν −¯ g ¯µν +∇ω TD D,X ∂µ φ∂ν φ∂ ω φ δφ + ∇ω TD D∂µ φδνω + TD ∂ν δµω δφ 2 2 2   √ √ −¯ g ¯µν −¯ g ¯µν −∇ω TD D∂µ φδνω + TD ∂ν δµω δφ . 2 2 (D.6) According to Stokes’ Theorem, and given that the fields vanish at the boundary (infinity), the terms corresponding to total divergences vanish and it is possible to write:  √ √ Z −¯ g ¯µν −¯ g ¯µν ω 4 TD (C,φ gµν + D,φ ∂µ φ∂ν φ) + ∇ω TD D,X ∂µ φ∂ν φ∂ φ δSD = d x 2 2 √  √  −¯ g ¯µν −¯ g ¯µν −∇µ TD D∂ν φ − ∇ν TD D∂µ φ δφ 2 2

(D.7)

√ At last, considering ∇ω ( −g) = 0 and taking T¯µν to be symmetric (which is consistent with symmetry in gµν ) we reach the final expression √ √  √ Z −¯ g ¯µν −g −¯ g ¯µν 4 ω δSD = d x TD (C,φ gµν + D,φ ∂µ φ∂ν φ) + ∇ω √ TD D,X ∂µ φ∂ν φ∂ φ 2 2 −g √  Z √ √ −¯ g ¯µν − −g∇µ √ TD D∂ν φ δφ ≡ d4 x −g Q δφ. −g

(D.8)

Combining equations (D.1), (D.4) and (D.8), renders the equation of motion for the scalar field: φ = V,φ − Q.

104

(D.9)

Appendix E

Disformally Coupled Quintessence Interaction term Let us consider the system with a canonical scalar field disformally coupled to a barotropic perfect fluid. For the background cosmology we assume a homogeneous, isotropic, spatially flat FLRW metric in cartesian coordinates, as described in (1.29), in the Einstein frame. The scalar field is assumed to depend only on time and coordinate time derivatives in this frame are, consistently, denoted by an upper dot. In these conditions, it is possible to rewrite the interaction term (4.15) as:

  1 2C 3 + C 2 (D,X − 2D) ρD φ˙ 2 + CD,X (D − 2C) ρD φ˙ 4 + CD,X − D,X ρD + 2D φ˙ 6 2  1 2 1 2 ˙ 10 + D,X (C − DρD ) φ˙ 8 − DD,X φ Q = −C 2 C,φ (ρD − 3pD ) + 6HC 2 DpD φ˙ 2 2 3 − C (CD,φ − 2DC,φ ) ρD φ˙ 2 − C 2 D,Xφ ρD φ˙ 4 + CC,φ D,X (ρD − 2pD ) φ˙ 4  2 1 3 2 − 6HCDD,X pD φ˙ 5 − C DD,Xφ − D,X D,φ ρD φ˙ 6 + C,φ D,X pD φ˙ 8 2 4   1 3 1 4 2 9 2 2 2 ˙6 2 ˙8 ˙ ˙ ˙ + HDD,X pD φ + C D,X φ + CDD,X φ − CD,X φ − DD,X φ ρD V,φ 2 2 2   1 1 2 ˙7 2 ˙9 + −C 2 D,X φ˙ 3 − CDD,X φ˙ 5 + CD,X φ + DD,X φ ρ˙ D 2 2   1 2 2 2 4 6 2 ˙8 ˙ ˙ ˙ ¨ − 2C D + 4C D,X φ + C (CD,XX + 2DD,X ) φ + CDD,XX φ + DD,X φ ρD φ. 2



(E.1)

This expression can be further simplified by using the modified Klein-Gordon equation, (4.23), and the fluid conservation equation, (4.24), to replace φ¨ and ρ˙ D in terms of the coupling Q: 105

   C,φ D,X X 1 C,φ D ˙ Q=G V,φ ρD + 3H (ρD + pD ) φ + 2ρD + 3 (3pD − ρD ) + (ρD − 2pD ) 2 C C C D  D,φ D,Xφ D,X X  ˙ − ρD X − 2 ρD X + 3H (5ρD + pD ) φ + 5ρD V,φ D D D      D,XX X 2  D 2 D,X X h ˙ + 6HρD φ + 2ρD V,φ + 6V,φ ρD X + 6H (3ρD − pD ) X φ˙ D C D   C,φ D,φ D,X X +2 −6H (ρD + pD ) X φ˙ + 6 ρD X 2 + pD X 2 − 2V,φ ρD X) D D C   2 D,Xφ X D,XX X 2 ˙ ρD X + −4 12H ρD X φ + 4V,φ ρD X D D (E.2) −1



with   D,X X D,XX X 2 D ρD − 2X + G ≡1 + (5ρD − 6X) + 2 ρD C D D #   2 "  D,XX X 2 D,X X D,X X 2 D (8X − 2ρD ) X + 4 + (6ρD + 4X) X + ρD X C D D D

106

(E.3)

Appendix F

Fixed Points for the Disformally Coupled System with no Kinetic Dependence Here we consider the system with a canonical scalar field disformally coupled to a barotropic perfect fluid, in the case where the disformal function only depends on the field. This corresponds to setting µ = 0 in the system where the disformal function can also depend on the kinetic term, given in equations (4.33), (4.34) and (4.36). The fixed points for a fluid with an arbitrary constant equation of state parameter γ are listed in Table F.1 and they were fully studied in [63]. Table F.1: Fixed points of the system (4.33)-(4.36) for the case µ = 0. Name

x

y

σ

(1) (2)

−1 1

0 0

0 0

q

(3) (4) (5) (6) (7) (8)

2 α(4−3γ) 3

γ−2 √λ q6 3 γ 2 (4−3γ)α+λ √ √ 2β− 2β 2 −3 √ 3 √ √ 2β+ 2β 2 −3 √ 3 q 3 γ 2

0 q √

1−

0 λ2

2α2 (4−3γ)2 +α(8−6γ)λ−3(γ−2)γ

√

0

0  p 1 2 − 6) − 3 2β(2β + 4β 18   p 1 2 4β − 6) − 3 18 2β(2β −

0

A

2(α(4−3γ)+λ)2

α(4−3γ)+2β

0

6



0

For the fixed point (8) in Table G.1 we have: A=

(α(4 − 3γ) + 2β)2 (2α2 (4 − 3γ)2 + 4αβ(4 − 3γ) − 3(γ − 2)γ) 9(γ − 1)γ(3(6α2 − 1)γ 2 − 24αγ(2α + β) + 8(2α + β)2 )

107

108

Appendix G

Disformally Coupled Quintessence Eigenvalues of the Stability Matrix Here we focus on the case of a canonical scalar field disformally coupled to a barotropic perfect fluid, as described in chapter 4. The naming of the fixed points follows the ones presented in Table 3.2 for the case of a non-relativistic fluid and in Table 4.3 for a relativistic fluid. According to what was described in section 1.1.1, the eigenvalues of the 3 × 3 linear stability matrix M, evaluated at each fixed point (xf , yf , σf ), are used as a tool to infer the stability of the fixed points of the system. The linear stability matrix is constructed from the dynamical system of equations (4.33)-(4.36). We choose to not present the eigenvalues for the disformal fixed points (σ 6= 0) due to their intricate analytical expression. The stability character of the disformal fixed points is studied numerically. • Eigenvalues for the case of a coupled relativistic fluid with equation of state parameter γ = 1 and with σ = 0: Table G.1: Eigenvalues for the fixed points of the system (4.33)-(4.36) for the case γ = 1 given in terms of the parameters. e1

Name (A+ )d (A− )d (B)d (C) (D)d

3+

√

6α

√ 3 − 6α − 32 + α2 −3 + 12 λ2

−3(2α+λ)−B 4(α+λ)

e2 q

e3

3 + 32 λ 3 2 2 + α + αλ −3 + λ(α + λ)

√ 2( 6β − 3(1 + µ)) √ −2( 6β + 3(1 + µ)) −4αβ − (3 + 2α2 )(1 + µ) λ(2β − λ(1 + µ))

−3(2α+λ)+B 4(α+λ)

3(2β−λ(1+µ) α+λ

3−

3 λ q2

For the fixed point (D)d in Table G.1 we have: B=

p −3(−72 − 12α(5α + 3λ) + 16αλ(α + λ)2 + 21λ2 ) 109

. • Eigenvalues for the case of a coupled relativistic fluid with equation of state parameter γ = 4/3, µ = 1 and σ = 0: Table G.2: Eigenvalues for the fixed points of the system (4.33)-(4.36) for the case γ = 4/3 and µ = 1 given in terms of the parameters. Name

e1

e2

e3

(O)r

2

−1 √
−2 6β + 6 2  q−4 + λ 1 64 − 15 − 1 2 λ2

q −8 3 2λ + 3 2λ(β − λ)

(A± )r (B)r (C)r

− 12

2 −3 + 12 λ2 q 64 λ2

− 15 + 1



110

8β λ

−8