Covariant Effective Actions in f(r) gravity for Modified Loop Quantum Cosmologies via Order Reduction

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UNIVERSIDADE DE LISBOA FACULDADE DE CI ˆENCIAS DEPARTAMENTO DE F´ISICA

Covariant Effective Actions in f (R) gravity for

Modified Loop Quantum Cosmologies via Order

Reduction

Ana Rita Lopes Ribeiro

MESTRADO EM F´ISICA

Especializac¸˜ao em Astrof´ısica e Cosmologia

Dissertac¸˜ao orientada por: Dr. Daniele Vernieri Dr. Francisco Lobo

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”You should be afraid of taking risks and pursuing something meaningful, but you should be more afraid of staying where you are if it’s making you miserable.” Jordan Peterson

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Ackowledgments

First and foremost, I would like to thank my supervisor, Dr. Daniele Vernieri for providing such an interesting topic for a thesis, in a field I was looking foward to learn more about, and for the advice he has given me, during this work. I would also like to thank my co-supervisor, Dr. Francisco Lobo, for his feedback and kind words.

I want to thank my parents for supporting me from the very begining. Even though we knew that pursuing the field of physics was a risky decision, especially since I was already enrolled in the field of biomedical engineering at the time, they encouraged me to pursue something that genuinely sparked my curiosity. Thank you for always wanting the best for me and for ensuring I could pursue a higher education. For the initial spark, I have to thank Dra. Ana Arriaga for the amazing lectures on modern physics, which inspired me to pursue the field of physics. I would also like to thank the excellent teachers I was fortunate to encounter throughout my academic path.

I want to thank my friends and family for their curiosity in my work and for asking questions. To my long time friends from high school, thank you for always being there throughout this journey, to share your stories and to listen to mine, even though we all chose different paths. A special thanks to my dear friend Catarina, who offered me, this past summer, Stephen Hawking’s book, Theory of Everything, because it reminded her of me. This book surely brought me inspiration to write this thesis. I want to also thank my dear college friends, Inˆes and Nichal, for always making my days brighter at University. To the good friends I made during this year, Daniel and Ricardo, thank you for the endless fun and online video game sessions, which helped maintain my mental sanity, during this bizarre year. Also, thank you Daniel, for helping me present the Mathematica graphs in more a understandable way, as well as with LATEX, regarding the presentation of equations. To Jo˜ao Maria, thank you for helping me review all of my

work, in the final stage. Thank you for your patience, for always being there for me and for inspiring me to be a better person. Last, but not least, I want to thank my cat, Oliver, for always keeping me company and relieving some of my stress.

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Abstract

The ΛCDM model relies upon the fact that General Relativity is the correct theory for gravity, concerning the large scales of the Universe. In fact, General Relativity is an extremely successful theory, as it was able to perfectly explain the precession of Mercury’s orbit or the gravitational deflection of light by the Sun, for example. The ΛCDM model itself is a very successful model for the Universe. Among other astrophysical and cosmological observations, it is able to explain the Cosmic Microwave Background and the large scale structure of the Universe.

Despite this, just as Newton’s theory of gravity has its limitations, and is only valid in a certain regime, the same is true for General Relativity. Extrapolating the expansion of the Universe backwards in time, using General Relativity, yields an infinite energy density, since all the mass is concentrated in a single point. This is the initial singularity problem, a point in time in which General Relativity is no longer valid. From this point on, the theory breaks down and the equations cannot be used to determine what happened before the Big Bang.

Quantum Gravity, a different description of gravity, is expected to provide more physical insight concerning this open question, since the problem lies on the dynamics of the Universe at very small scales, at the order of the Planck length. In fact, one alternative scenario to the Big Bang, that manages to completely avoid the singularity, is offered by Loop Quantum Cosmology, a reduced model of Loop Quantum Gravity, which predicts that the Universe evolves in a cyclic manner, going from collapse to expansion through a bounce.

In this thesis, we use metric f (R) gravity, a modification of General Relativity, to reproduce the modified Friedmann equations, which are obtained in different models of Loop Quantum Cosmology. To achieve this, we applied an order reduction method to the f (R) field equations, having obtained covariant effective actions that lead to a bounce, for mLQC-I and mLQC-II, considering matter as scalar field, as well as for a more general scenario.

Keywords: initial singularity, cyclic cosmology, loop quantum cosmology, f (R) gravity, covariant order reduction

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Resumo

O modelo ΛCDM assume que a teoria da Relatividade Geral ´e a correcta teoria que descreve a gravidade, no que diz respeito `as grandes escalas do Universo. De facto, esta teoria ´e extremamente bem sucedida, tendo sido capaz de explicar a precess˜ao da ´orbita de Merc´urio ou a deflex˜ao gravitacional da luz pelo Sol, por exemplo. O modelo ΛCDM ´e tamb´em, ele pr´oprio, um modelo bem sucedido para o Universo. Entre muitas outras observac¸˜oes astrof´ısicas e cosmol´ogicas, ´e capaz de explicar a Radiac¸˜ao C´osmica de fundo e a estrutura de larga escala do Universo.

Apesar dos factos mencionados, tal como a teoria da gravidade de Newton tem as suas limitac¸˜oes, e ´e apenas v´alida em certos regimes, o mesmo ´e verdade para a teoria da Relatividade Geral. Extrapolando a expans˜ao do Universo para tr´as no tempo, usando Relatividade Geral, origina uma densidade infinita, uma vez que toda a massa fica concentrada num ´unico ponto. Este ´e o problema da singularidade inicial, um ponto no tempo no qual a Relatividade Geral j´a n˜ao ´e v´alida. A partir deste ponto, a teoria deixa de funcionar e as suas equac¸˜oes n˜ao podem ser utilizadas para determinar o que aconteceu antes do Big Bang.

Espera-se que a Gravidade Quˆantica, uma descric¸˜ao diferente da gravidade, fornec¸a mais informac¸˜ao sobre esta quest˜ao em aberto, uma vez que o problema se encontra na dinˆamica do Universo em escalas muito pequenas, na ordem do comprimento de Planck. De facto, um cen´ario alternativo ao Big Bang, que ´e capaz de evitar a existˆencia de uma singularidade por completo, ´e oferecido pelo ramo da Cosmologia Quˆantica em Loop, um modelo reduzido da Gravidade Quˆantica em Loop, que prevˆe que o Universo evolui de uma fase de colapso para uma fase de expans˜ao atrav´es de um ”salto”, fazendo-o de forma c´ıclica.

Nesta dissertac¸˜ao, iremos usar o formalismo da m´etrica da teoria f (R) da gravidade, uma modificac¸˜ao `a Relatividade Geral, para reproduzir as equac¸˜oes modificadas de Friedmann, que s˜ao obtidas em difer-entes modelos de Cosmologia Quˆantica em Loop. Para isso, recorremos ao m´etodo de reduc¸˜ao de or-dem, aplicando-o `as equac¸˜oes de campo da teoria f (R), tendo obtido acc¸˜oes covariantes efectivas que descrevem um Universo c´ıclico, de acordo com mLQC-I e mLQC-II, considerando a mat´eria como um campo escalar, assim como o caso mais geral .

Palavras-chave: singularidade inicial, cosmologia c´ıclica, cosmologia quˆantica em loop, gravidade f (R), reduc¸˜ao de ordem covariante

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Contents

Contents vii

List of Figures ix

List of Abbreviations x

1 Introduction 1

1.1 ΛCDM model and inflation . . . 1

1.2 Initial singularity problem . . . 2

1.3 Modified theories of gravity and f (R) gravity . . . 3

1.3.1 Bouncing cosmology in metric f (R) gravity . . . 4

1.4 Thesis Outline . . . 5

2 General relativity and f (R) gravity 7 2.1 General relativity . . . 7

2.1.1 Geometric description of space-time . . . 7

2.1.2 Lagrangian formulation of general relativity . . . 12

2.1.3 Friedmann and Raychaudhuri equations . . . 15

2.2 f (R) gravity . . . 17

2.2.1 Lagrangian formulation of metric f (R) gravity . . . 18

2.2.2 Generalized Friedmann and Raychaudhuri equations . . . 21

3 Loop quantum cosmology and its modifications 23 3.1 Loop quantum gravity and loop quantum cosmology . . . 23

3.2 Modifications of loop quantum cosmology . . . 26

3.2.1 Modified loop quantum cosmology - I . . . 26

3.2.2 Modified loop quantum cosmology - II . . . 29

4 Covariant order reduction technique 31 4.1 Theoretical framework . . . 31

4.2 Reduced equations in f (R) gravity . . . 33

4.2.1 Time-time component of the reduced equations . . . 33

5 Effective actions for loop quantum cosmology models 35 5.1 Matter as a scalar field . . . 36

5.1.1 Effective action for loop quantum cosmology . . . 37

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5.1.3 Effective action for modified loop quantum cosmology II . . . 41

5.1.4 Graphic representation of the solutions . . . 44

5.2 General w . . . 48

5.2.1 Effective action for loop quantum cosmology . . . 48

5.2.2 Effective actions for modified loop quantum cosmology I . . . 49

5.2.3 Effective action for modified loop quantum cosmology II . . . 53

6 Conclusions and future perspectives 55 Bibliography 59 Appendix A Mathematica codes for the b−branch of mLQC-I 63 A.1 Matter as a scalar field . . . 63

A.1.1 Solving the differential equation . . . 63

A.1.2 Simplifying the solution . . . 64

A.1.3 Obtaining a series expansion . . . 65

A.2 General w case . . . 65

A.2.1 Solving the differential equation . . . 65

A.2.2 Simplifying the solution . . . 66

A.2.3 Obtaining a series expansion . . . 68

A.2.4 Recovering the w = 1 result . . . 68

Appendix B Mathematica codes for the b+branch mLQC-I 71 B.1 Matter as a scalar field . . . 71

B.1.1 Solving the differential equation . . . 71

B.1.2 Simplifying the solution . . . 72

B.1.3 Obtaining a series expansion . . . 73

B.2 General w case . . . 73

B.2.1 Solving the differential equation . . . 74

B.2.2 Simplifying the solution . . . 75

B.2.3 Obtaining a series expansion . . . 75

B.2.4 Recovering the w = 1 result . . . 76

Appendix C Mathematica codes for mLQC-II 79 C.1 Matter as a scalar field . . . 79

C.1.1 Solving the differential equation . . . 79

C.1.2 Simplifying the solution . . . 80

C.1.3 Obtaining a series expansion . . . 81

C.2 General w case . . . 81

C.2.1 Solving the differential equation . . . 82

C.2.2 Simplifying the solution . . . 83

C.2.3 Obtaining a series expansion . . . 84

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List of Figures

5.1 Deviation, from GR, of the effective lagrangian: a comparison between the bbb−−−

branch of mLQC-I and LQC. This graph presents the comparison between eq. (5.27), for the b−branch of mLQC-I (dashed dark orange line), and eq. (5.21), for LQC (solid

blue line), with c1 = c2 = 0, for −2κρIc < R < 0. Although the deviation is higher

for the case of the b−branch of this modified model, both of them are dominated by the

quadratic term. . . 44 5.2 Effective f (R)f (R)f (R) function: a comparison between the bbb−−− branch of mLQC-I, LQC

and GR. This graph illustrates the differences between the lagrangian density of GR (solid black line), given by eq. (2.40) with Λ = 0, and the effective lagrangian densities given by eq. (5.22), in the case of LQC (solid blue line) and eq. (5.31), for the b−branch

of mLQC-I (dashed dark orange line), for the interval −2κρIc < R < 0. . . 44 5.3 Deviation, from GR, of the effective lagrangian: a comparison between the bbb+++

branch of mLQC-I and LQC. This graph presents the comparison between eq. (5.37), for the b+branch of mLQC-I (dashed light orange line), and eq. (5.21), for LQC (solid

blue line), with c1 = c2 = 0, for −2κρIc < R < 0. Contrary to the previous case, for

this branch the correction is dominated by the linear term. As a result, since R < 0, the deviation is predominantly negative. . . 45 5.4 Effective f (R)f (R)f (R) function: a comparison between the bbb+++ branch of mLQC-I, LQC

and GR. This graph illustrates the differences between the lagrangian density of GR (solid black line), given by eq. (2.40) with Λ = 0, and the effective lagrangian densities given by eq. (5.22), in the case of LQC (solid blue line), and eq. (5.31), for the b+branch

of mLQC-I (dashed light orange line), for the interval −2κρIc < R < 0. . . 45 5.5 Deviation, from GR, of the effective lagrangian: a comparison between

mLQC-II and LQC. This graph presents the comparison between eq. (5.44), for the case of mLQC-II (dashed pink line), and eq. (5.21), for LQC (solid blue line), with c1= c2= 0,

for −18κρc< R < 0. Both of the corrections are dominated by the quadratic term. . . . 46

5.6 Effective f (R)f (R)f (R) function: a comparison between mLQC-II, LQC and GR. This graph illustrates the differences between the lagrangian density of GR (solid black line), given by eq. (2.40) with Λ = 0, and the effective lagrangian densities given by eq. (5.22), in the case of LQC (solid blue line), and eq. (5.46), for mLQC-II (dashed pink line), for the interval −18κρc< R < 0. . . 46

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5.7 Deviation, from GR, of the effective lagrangian: a comparison between all models. This graph contains eq. (5.21), for LQC (solid blue line), eq. (5.27), for the b−branch

of mLQC-I (dashed dark orange line), eq. (5.37), for the b+branch of mLQC-I (dashed

light orange line) and eq. (5.44), for mLQC-II (dashed pink line), with c1 = c2 = 0, for

−2κρI

c < R < 0. . . 47

5.8 Effective f (R)f (R)f (R) function: a comparison between all models. This graph presents the effective f (R) functions for all models, including GR, given by eq. (2.40) with Λ = 0. These are eq. (5.22), for LQC (solid blue line), eq. (5.31), for the b−branch of mLQC-I

(dashed dark orange line), eq. (5.39), for b+ branch of mLQC-I (dashed light orange

line) and eq. (5.46), for mLQC-II (dashed pink line), for −2κρIc < R < 0. . . 47 5.9 Deviation, from GR, of the effective lagrangian: a comparison between the solution

for matter as a scalar field and the one we get from the general solution, for the b+

branch of mLQC-I. This graph contains the solution for matter as scalar field (dashed light orange line) that we have obtained in the previous section, given by eq. (5.37), and the solution we get when substituting w = 1 in the general solution (solid blue line), for −2κρI

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List of Abbreviations

CMBR . . . Cosmic Microwave Background Radiation CDM . . . Cold Dark Matter

GR . . . General Relativity IR . . . Infrared

UV . . . Ultraviolet

QFT . . . Quantum Field Theory E-H . . . Einstein-Hilbert

FLRW . . . Friedmann-Lemaˆıtre-Robertson-Walker EoS . . . Equation of State

LQG . . . Loop Quantum Gravity LQC . . . Loop Quantum Cosmology

mLQCs . . . modified Loop Quantum Cosmologies mLQC-I . . . modified Loop Quantum Cosmology I mLQC-II . . . modified Loop Quantum Cosmology II

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Chapter 1

Introduction

1.1

ΛCDM model and inflation

Our latest observations, coming from different astrophysical and cosmological sources, such as the CMBR and supernovae surveys, seem to indicate that our Universe is composed of ordinary baryonic matter (4%), dark matter (20%) and mainly composed of an unknown form of energy (76%), dubbed dark energy [1]-[3]. Moreover, the discovery of the CMBR, is a key observation that supports the idea of a primordial hot and dense Universe, that has been expanding ever since. The standard model of cosmology, that adequately fits the current available data is the ΛCDM model, sometimes also called concordance model, supplemented with an inflationary scenario.

In ΛCDM, Λ stands for a cosmological constant and CDM stands for Cold Dark Matter. The presence of a cosmological constant in this model is due to the fact that it is the simplest explanation for dark energy, being an intrinsic energy of the Universe responsible for its current accelerated expansion. Dark energy is also characterized by the fact that it does not exhibit the clustering properties of regular matter. Cold dark matter, on the other hand, refers to an unknown form of matter and is the standard answer to the ”missing mass problem” in astrophysics that is summed up by the fact that our current understanding of the dynamics of galaxies is incomplete. This problem was first adressed for galaxy clusters in [4] and for individual galaxies in [5], however a satisfactory final answer is yet to be found.

The ΛCDM model is complemented with an inflationary epoch, that is needed to adress the horizon and the flatness problems, as well as the monopole problem, which we will not be discussing here. The horizon problem has to do with the observed homogeneity of causally disconnected regions of the Universe. We can observe this in the CMBR, where the fluctuations in temperature are extremely small. The flatness problem is related with the value of the matter density at the present time, which is very close to the critical value that allows for a spatially flat space-time and which corresponds to the borderline between an open and closed Universe. Moreover, its initial value must have already been very much close to the critical value, and any deviation of this initial condition would have been amplified, leading to a non-flat Universe today. Both of these problems can be viewed as fine-tuning problems, in the sense that the initial conditions of the Universe seem to have been very specific to allow for the Universe we observe today. Here, we clarify that by initial conditions we do not mean the conditions precisely at the beginning of time, since this model has a singularity at t = 0, which we will discuss in a moment. It is, therefore, sensible to set initial conditions only after quantum gravitational effects stop being essential. Physicist Alan Guth adressed these issues in [6], arguing that the Universe experienced an extreme accelerated expansion, in a short period of time, after the Big Bang, which was named inflation. With respect to the horizon problem, the inflationary scenario is a possible solution since, prior to this extreme expansion,

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the observable Universe we see today was small enough to be causally connected. Inflation then allowed for the Universe to grow with an unprecedented rate, in a comparatively short amount of time, giving rise to the causally disconnected regions we observe today. Moreover, even a globally non-flat Universe is locally flat and, in this sense, it seems that inflation was also responsible to amplify these smaller flat regions. In this picture, the Universe we currently observe is just a small part of the entire Universe that itself may be curved. This is how inflation also possibly solves the flatness problem.

Assuming GR as the correct theory of gravity, the ΛCDM model makes use of the Friedmann and Raychaudhuri equations to describe the evolution of a homogeneous and isotropic Universe, explaining its dynamics and the structures it contains. Nevertheless, this model and, more generally, GR still suffer from many open questions, which can be divided in two main categories, IR scales and UV scales, that is at cosmological/astrophysical scales and quantum scales [7]-[8].

Open questions at IR scales include those related with dark matter and dark energy. For example, the ΛCDM model is not able to explain the nature of dark matter by itself. There are also problems related to the cosmological constant, namely the magnitude and coincidence problems. The first one has to do with the fact that the observed value for the cosmological constant is too small compared to the one we find in QFT. The coincidence problem, on the other hand, is related with the short period of time in the history of the Universe in which the energy density of the cosmological constant is comparable with that of matter.

UV scales’ shortcomings, on the other side, refer to those related with the quantum gravity problem. GR and quantum mechanics, although very successful in their own areas, are incompatible when both are needed to describe certain physical scenarios, such as the presence of a strong gravitational field at quantum scales. On one hand, QFT assumes that space-time is flat and even QFT in curved space-time assumes that such space-time is a rigid arena. GR, on the other, simply does not take into account the quantum nature of matter, describes space-time as a dynamical entity and is not a renormalizable theory. As such there has been an effort to put gravity on the same footing as the other interactions. A particular example of a scenario in which a quantum representation of gravity seems to be needed is the Big Bang scenario in which the Universe dimensions inevitably became smaller than the Planck scale, so much so that, according to the standard model of cosmology, the Universe had infinite density. This is called the initial singularity problem.

1.2

Initial singularity problem

When extrapolated into the past, the ΛCDM model exhibits a singularity, which we call the Big Bang singularity. The Big Bang is not a physical event, but a limit to our theoretical model, for it represents the moment when the Universe energy density and space-time curvature were infinite. At this point, our equations are no longer valid, making it impossible for us to predict what happened before the Big Bang. In 1965, Roger Penrose formulated a theorem, using the behavior of light cones and the fact that gravity is always atractive, to show that a massive collapsing star becomes confined in a region whose surface decreases until it reaches zero. This means that its mass is compressed into a region of zero volume and the density and space-time curvature become infinite, forming a singularity known as a black hole. Stephen Hawking, who was a PhD student at the time, questioned if a Big Bang singularity was really necessary in the framework of GR. Hawking realized that if time was inverted in the Penrose theorem, transforming the contraction into an expansion, the conditions of the theorem remained verified, as long as the Universe, today and on large scales, resembles the Friedmann model. For tecnhical reasons, the Penrose theorem demanded that the Universe was infinite in size. Thus, in his PhD thesis, Hawking

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was able to prove that the singularity should only exist if the Universe was expanding fast enough to prevent a contraction, since this is the only Friedmann model that is infinite in space [9], [10]. In 1970, Penrose and Hawking developed new mathematical tools to remove this and other conditions, proving the need for a Big Bang singularity, as long as GR is the correct theory of gravity and the Universe contains as much matter as we currently observe [9], [11].

At this point, we emphasize that the conclusion that the Universe appears to have emerged from a singularity is based on purely classical considerations. A theory of gravity in the quantum regime is expected to provide a more complete understanding of the Big Bang, and maybe even remove the initial singularity. One such possibility is offered by loop quantum cosmology which, through an effective Hamiltonian description, leads to an effective Friedmann equation [12], given by

H2 = 1 3κρ  1 − ρ ρc  , (1.1)

which, as we can observe, yields a quantum bounce instead of a Big Bang, when the energy density reaches the critical value ρc. Similar results were obtained by two modified models of loop quantum

cosmology, in which the Big Bang is replaced by a quantum bounce. These will be discussed in chapter 3.

1.3

Modified theories of gravity and f (R) gravity

In the context of GR, one approach towards resolving the mentioned problems and, in particular, the initial singularity problem, is to consider our description of gravity as insuficient, at the relevant scales. In this sense, several modified theories of gravity appeared over the course of time, most of them driven by the discovery of dark matter and dark energy. A few examples of reviews are given by [8], [13] -[15], where recent work on modified/extended theories of gravity and their cosmological consequences are presented as well as their theoretical foundations and experimental tests of each class of theories. From a theoretical point of view, pursuing modified theories of gravity is important because it enables us to understand the nature of GR and the cosmological consequences of moving away from it. In doing so, it might be possible to find answers to the IR shortcomings of GR, and maybe even to give an effective description of Quantum Gravity, while not spoiling the already successful predictions of standard cosmology.

First attempts at deviating from GR appeared few years after its foundation (1915). As there was no well-posed motivation to change it, these were mainly triggered by scientific curiosity. The first real motivations appeared in the 1960’s and were concerned with the fact that GR is not renormalizable and, therefore, cannot be quantized in the traditional way. From this angle, in 1962, Utiyama and De Witt showed that it is necessary for the E-H action to have higher-order curvature terms for renormalization, at one-loop, to be possible [16]. Later on, Stelle showed that higher-order actions are indeed renormalizable [17]. As we will discuss on chapter 2, the E-H action, which is given by

SEH = 1 2κ Z V d4x√−g(R + 2Λ) + SM(gµν, Ψ ) , (1.2)

is the covariant action that leads to the Einstein field equations, and its Lagrangian is the Ricci scalar, R, with the addition of a cosmological constant Λ. One way to add such higher-order curvature invariants in the E-H action, is by making a straightforward generalization of its Lagrangian, such as replacing the

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Ricci scalar by a general function of it S = 1 2κ Z V d4x√−gf (R) + SM(gµν, Ψ ) , (1.3)

where, viewed as a series expansion, f (R) = ... + α2 R2 + α1 R + 2Λ + R + R2 β2 +R 3 β3 + ... , (1.4)

where αi and βi are coefficients with the appropriate dimensions. This specific modification is what

caracterizes the class of f (R) modified theories of gravity. In this work we will be considering the metric formalism of f (R) gravity and, in chapter 2, we describe the process by which we get the corresponding field equations.

The addition of higher-order curvature terms was first considered to be important only at scales close to the Planck scale, namely in the early Universe or near black hole singularities. In fact, the class of f (R) modified theories of gravity can lead to a period of accelerated expansion in the early Universe if quadratic corrections are added to the E-H Lagrangian, which is originally called the Starobinsky model [18]. More recently, a selection of possible inflationary scenarios was presented in the Planck data releases and it was found that the Starobinsky model is the best fit to the available data [19]-[20].

However, it was latter verified that f (R) gravity can also lead to late-time accelerated expansion without the need for dark energy. By defining an effective energy density ρef f, effective pressure pef f

and, as a result, an effective EoS parameter, one is able to write the field equations, that come from f (R) gravity, with the form of the standard Friedmann equations, where the curvature correction can be interpreted as an effective fluid [21]. Moreover, assuming a power law solution for the scale factor, it was showned that a function of the form f (R) ∝ R2gives rise to accelerated behaviours with the correct EoS parameter wef f = −1 [22]. There have also been attemps to use metric f (R) as a substitute for

dark matter, at galactic and cluster scales. Motivated by the fact that dark matter has not been detected yet, some authors considered the possibility that the form of galaxy rotation curves is not an indication of the existence of dark matter halos, but an indication of the need for different gravitational physics, for instance in [23].

More general Lagrangians, such as f (R, RµνRµν, RµναβRµναβ), which include other higher-order

invariants could be considered, although we have reasons not to. First of all, f (R) modified theories of gravity are sufficiently general to encapsulate some of the basic characteristics of higher-order gravity and, at the same time, they are simple enough to be handled in terms of calculations, compared to other modified theories of the same type. These theories also seem to be the only ones which can avoid the Ostrogradski instability1[13], [24].

1.3.1 Bouncing cosmology in metric f (R) gravity

With all the previous aspects taken into account, a metric f (R) action, with a Lagrangian given by f (R) = R + 1

18κρc

R2+ ... , (1.5)

1There is a linear instability in the Hamiltonian associated with Lagrangians which depend upon more than one time

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was found which, when treated as an effective action, leads to the effective Friedmann eq. (1.1) [25]. In chapter 5, we discuss the procedure that leads to this effective action. The same result could have been obtained if one followed the same procedure for an analytic function f (R, RµνRµν, RµναβRµναβ)

instead of simply f (R). Moreover, due to the Gauss-Bonnet theorem, the family of Lagrangians of the form

f (R, RµνRµν) = R + aR2+ bRµνRµν (1.6)

lead to the same field equations as the Lagrangian

f (R) = R + cR2 , (1.7)

with a + b/3 = c. In the case of LQC, the family of Lagrangians of the form of eq. (1.6), with a + b/3 = 1/(18κρc), also lead to the effective Friedmann eq. (1.1) [26].

1.4

Thesis Outline

In this thesis, our goal is to modify GR, using metric f (R) gravity, in such a way that we are able to find effective actions, that lead to the modified Friedmann equations derived in the field of LQC, all of which yield a quantum bounce.

The structure of this thesis is the following. In chapter 2, the essential geometric description of space-time in the framework of GR is given, which is then used to formulate GR in a Lagrangian point of view. From the tensorial expression of the equations that we obtain, the Friedmann and Raychaudhuri equations are derived, in the context of a homogeneous and isotropic space-time, by considering a FLRW metric. We then consider a modification of the Lagrangian, through metric f (R) gravity and derive, in detail, the new field equations. Once again, considering a FLRW metric, we derive the modified Friedmann and Raychaudhuri equations, in the context of f (R) gravity.

In chapter 3, we motivate the need for a quantum theory of gravity and explore, in a brief manner, one of the possibilities, loop quantum gravity. We then delve into loop quantum cosmology, the symmetry-reduced model of loop quantum gravity, and show how in this framework we get an equation, for the evolution of the Universe, that is just the usual Friedmann equation with a modified source, that gives rise to a quantum bounce, rather than a Big Bang. Moreover, we also discuss the need for modified models of loop quantum cosmology and focus on two specific models, describing the derivation of their corresponding dynamical equations.

In chapter 4, we adress the fact that the modified field equations, that we obtained in chapter 2 in the context f (R), are higher than second order differential equations, and explain why this is an issue. We motivate and describe the method of covariant order reduction, that allows us to get second order equations, that give solutions perturbatively close to GR. We apply this method in our modified field equations, obtaining a reduced version of the modified Friedmann equation.

In chapter 5, we present our results. We devide this chapter in two sections, considering matter as a scalar field first, by fixing the EoS parameter to w = 1, and then generalize the results for general w. In each of these sections, using the reduced version of the modified Friedmann equation, we derive covariant effective actions, for each of the mLQC models that are being considered, namely mLQC-I and mLQC-II. With respect to standard LQC, we present the results of [25] and derive its corresponding generalization.

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Chapter 2

General relativity and f (R) gravity

2.1

General relativity

It took Albert Einstein over ten years to move from the formulation of special relativity, in 1905, to a final formulation of GR, during 1915. Through thought experiments, creative thinking and, most certaintly, hard work, Einstein constructed a more general theory, that provides a description of gravity as a geometric property of space and time, being one of the greatest achievements of the human mind.

2.1.1 Geometric description of space-time

In this section, we aim to describe the basic mathematical entities that appear in the framework of GR. We start with the basic formalism of tensors and basic tensor calculations, following the textbook [27].

In GR we use the language of tensors, which helps to summarize sets of equations succinctly and, as a consequence, to solve problems in a faster and easier way, while also revealing structure in the equations. Tensors are defined on a differential manifold. In simple terms, a manifold is something which locally looks like n-dimensional Euclidean space Rn. For example, a small portion of the surface of a sphere looks like a plane surface. An n-dimensional manifold is a set of points such that each point possesses a set of n coordinates (x1, x2, ..., xn), where each coordinate may range from −∞ to +∞. In GR we have a 4-dimensional manifold, and so, a point on the manifold has coordinates (x1, x2, x3, x4), the first one being the time coordinate and rest being the spatial coordinates.

We define two categories of tensors, contravariant and covariant tensors. We define them in terms of their transformation properties, under a coordinate transformation. A contravariant tensor, of rank 1, is a set of quantities, written Xν in the xν-coordinate system, which transforms under a change of coordinates according to

X0ν = ∂x

∂xµX

µ . (2.1)

where x0νare the coordinates in the new coordinate system. Here, we introduce the Einstein summation convention. In eq. (2.1) the index µ is repeated on the right-hand side of the equality, implying a summation, like so X0ν = 4 X µ=1 ∂x0ν ∂xµX µ , (2.2)

where the index ν can be set to be equal to 1, 2, 3 or 4 and has to be the same on both sides of the mathematical equality. The repeated index µ is usually called dummy index, since it can be replaced without changing the meaning of the expression. From this point on, we will be using this convention

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without referring to it. The other tensor category, the covariant tensor, is defined by the transformation Xν0 = ∂x

µ

∂x0νXµ , (2.3)

for the case of a rank 1 tensor. We can also define mixed tensors. For example, a mixed tensor of rank 3 - one contravariant rank and two covariant rank - satisfies the transformation

X0νµρ= ∂x 0ν ∂xη ∂xα ∂x0µ ∂xβ ∂x0ρX η αβ . (2.4)

The rank tells us how many indices the tensor has. In particular, a tensor of rank zero is a scalar, a tensor of rank 1 is a vector and a tensor of rank 2 is a matrix. The reason why tensors are important in mathematical physics is because a tensor equation that holds in one coordinate system necessarily holds in all coordinate systems or, in other words, tensorial equations are coordinate independent.

Covariant derivative and affine connection

We will start with the definition of covariant derivative. In order to do that, consider a contravariant vector field Xν evaluated at a point P with coordinates xν, Xν(x), and at a point Q with coordinates xν + δxν, Xν(x + δx). Consider also a vector at Q which is parallel to Xν at P . Since xν + δxν is close to xν, we can assume that the parallel vector only differs from Xν(x) by a small amount, which we denote δXν(x). As such, the vector at Q, that is parallel to the vector at P is given by Xν(x) + δXν(x). Then, by definition, the covariant derivative is given by the limiting case

∇µXν = lim δxµ→0 1 δxµX ν(x + δx) −Xν(x) + δXν(x) , (2.5)

where we have the difference between the vector already at Q, Xν(x + δx), and the vector, also at Q, that is parallel to the vector at P , Xν(x) + δXν(x). Given that δXν(x) should vanish whenever Xν(x) or δxν do, then we define it as directly proportional to both,

δXν(x) = −Γηµν (x)Xη(x)δxµ , (2.6)

where Γηµν (x) are mutiplicative factors (analogous to proportionality constants), called affine connection or simply connection. Using the Taylor expansion Xν(x + δx) = Xν+ δxη∂ηXν and eq. (2.6), we have

µXν = ∂µXν + Γηµν Xη . (2.7)

It can be shown, that covariant differentiation satisfies the Leibniz rule. Knowing this, we find

∇µXν = ∂µXν− Γνµη Xη . (2.8)

Finaly, for a general tensor Tρ...ν..., the covariant derivative is given by

µTρ...ν... = ∂µTρ...ν...+ Γηµν Tρ...η...+ ... − Γρµη Tη...ν...− ... . (2.9)

The covariant derivative of a scalar field is the same as its ordinary derivative, i.e. ∇µφ = ∂µφ and the

difference of two connections is a tensor called torsion tensor, Tηµν = Γηµν − Γν

µη, not to be confused with

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the connection is said to be symmetric. Affine geodesics

Consider a congruence of curves defined such that only one curve goes through each point in the man-ifold. A contravariant vector field X determines a local congruence of curves xα = xα(u), where the tangent vector field to the congruence is Xα = dxα/du. For any tensor, Tα...

β..., we define the absolute

derivative along a curve C of the congruence as D Du T

α...

β... = ∇XTβ...α... , (2.10)

where ∇X = Xη∇η. If the absolute derivative is zero, then Tβ...α...is said to be parallely propagated. An

affine geodesic is defined as a privileged curve along which the tangent vector is propagated parallel to itself, like so D Du  dxα du  = λ(u)dx α du ⇔ ∇XX α = λXα (2.11)

or equivalently, using eq. (2.7),

d2xα du2 + Γ α βγ dxβ du dxγ du = λ dxα du . (2.12)

If the curve is parametrized in such a way that λ vanishes, then the parameter is called an affine parameter and the affine geodesic equation reduces to

d2xα du2 + Γ α βγ dxβ du dxγ du = 0 . (2.13) Riemann tensor

In general, covariant differentiation is not commutative. For any tensor Tβ...α..., we define its commutator to be

∇ρ∇σTβ...α...− ∇σ∇ρTβ...α... . (2.14)

In the case of a vector Xµ, using eq. (2.9), its commutator is given by

∇ρ∇σXµ− ∇σ∇ρXµ= RµνρσXν + (Γρσe − Γσρe )∇eXµ , (2.15)

where Rµνρσis the Riemann tensor defined by

νρσ = ∂ρΓνσµ − ∂σΓνρµ + Γνσα Γαρµ − ΓνραΓασµ . (2.16)

For the case of torsion-free connections, i.e. symmetric connections Γρση = Γσρη , the commutator is

simply given by

ρσXµ− ∇σρXµ= RµνρσXν . (2.17) This relation is often called Ricci identity. It can be shown that, for a symmetric connection, the com-mutator of any tensor can be expressed in terms of the tensor itself and the Riemann tensor. Thus, the vanishing of the Riemann tensor is a necessary and sufficient condition for the vanishing of the commu-tator of any tensor. We can generalize the Ricci identity for a tensor of rank 2, as

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Metric tensor

Any symmetric covariant tensor field of rank 2, say gµν(x), defines a metric. A manifold endowed with a

metric is called a Riemannian manifold. A metric can be used to define distances and lengths of vectors. The infinitesimal interval, which we denote by ds, between two neighbouring points xν and xν+ dxν is defined by

ds2 = gµν(x)dxµdxν , (2.19)

which is also known as the line element. The determinant of the metric is denoted by g = det(gµν).

From linear algebra, we know that the metric is non-singular if g 6= 0, in which case the inverse of gµν,

gµν, is given by gµνgνρ = δρµ, where δρµis the Kronecker delta, which is always equal to zero, except

when ρ = µ, when it is equal to one. It follows from this definition that gµν is a contravariant tensor of rank 2 and is called the contravariant metric. We may now use gµν and gµν to lower and raise tensorial

indices by defining

T...µ...... ... = gµνT... ...ν... (2.20)

and

T... ......µ... = gµνT...ν...... ... . (2.21) In this context, we have to be careful about the order in which we write contravariant and covariant indices. For example, Xµν 6= Xν

µin general. Finally, the metric tensor can also be used to calculate the

trace of a tensor of rank 2, Tµν, which is defined by

T = gµνTµν . (2.22)

Metric geodesics

Consider a timelike curve C with parametric equation xα = xα(u). Using eq. (2.19), the interval s between two points P1and P2 on C is given by the integral

s = Z P2 P1 ds = Z P2 P1 ds dudu = Z P2 P1  gαβ dxα du dxβ du 12 du . (2.23)

A timelike metric geodesic is defined as the curve between points P1 and P2 whose interval is

sta-tionary, under small variations that vanish at the boundaries. The equations we obtain, from δs = 0, are called the Euler-Lagrange equations. Multiplying the result by gαη, and using the fact that gαβgαη= δβη,

we finally get the metric geodesic equations, which are given by d2xα du2 + nα βγ odxβ du dxγ du =  d2s du2  ds du  dxα du , (2.24)

where dummy indices have been renamed. The curly brackets nα

βγ

o

are called the Christoffel symbols of the second kind and are given by

nα βγ o = 1 2g αη(∂ γgβη+ ∂βgγη− ∂ηgβγ) . (2.25)

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geodesic become d2xα du2 + nα βγ odxβ du dxγ du = 0 . (2.26) Metric connection

In general, if we have a manifold endowed with both an affine connection, Γβγα, and a metric, gαβ,

then it possesses affine geodesics and metric geodesics, which will be different, in principle. However, comparing eq. (2.13) with eq. (2.26), the two classes will coincide if Γβγα =nα

βγ o , in which case Γβγα = 1 2g αη(∂ βgηγ+ ∂γgηβ− ∂ηgβγ) . (2.27)

In fact, this is the most basic assumption of GR. It follows from the last equation that, in this case, the connection is necessarily symmetric, Γβγα = Γγβα . This special connection, built out of the metric and its derivatives, is called the metric connection. Using the covariant derivative formula, given by eq. (2.9), this definition leads immediately to the identity

∇ρgµν = 0 . (2.28)

Curvature tensor

The curvature tensor, or Riemann-Christoffel tensor, is defined by eq. (2.16), where the connection Γβµα is the metric connection, which is given by eq. (2.27). Thus, Rαβµν depends on the metric and its first and second derivatives, satisfying

Rαβµν = −Rαβνµ= −Rβαµν= Rµναβ (2.29)

and

Rαβµν+ Rανβµ+ Rαµνβ = 0 . (2.30)

We can use the curvature tensor to define several other important tensors. For instance, the Ricci tensor is defined by the contraction

Rµν = Rαµαν = gασRσµαν . (2.31)

The trace is given by eq. (2.22). As such, a final contraction of the Ricci tensor, defines the curvature scalar or Ricci scalar R as

R = gµνRµν . (2.32)

These two tensors can be used to define the Einstein tensor as Gµν = Rµν−

1

2gµνR . (2.33)

Einstein field equations

The full field equations, which hold in the presence of fields other than gravitation, are given by

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where κ is the coupling constant, which is given by κ = 8πG

c4 , (2.35)

G is the gravitational constant and c is the speed of light. In this work, we will be using the geometrized system of natural units, in which

c = ~ = G = 1 , (2.36)

κ = 8π . (2.37)

The equivalence of mass and energy, from special relativity, already suggests that all forms of energy act as sources for the gravitational field. In GR, these fields are contained in the energy-momentum tensor, Tµν, describing all forms of energy in the universe. The Einstein tensor, on the other hand, describes

the geometry of space-time. In that regard, eq. (2.34) are famously summarized by John Wheeler with the following statement: matter, Tµν, tells space-time geometry, Gµν, how to curve and geometry tells

matter how to move.

The cosmological constant, Λ, was originally included in the equations by Einstein, in order to describe a static universe. This idea persisted until Friedmann proposed another view, as we will discuss later on.

2.1.2 Lagrangian formulation of general relativity

This formulation of GR was developed by David Hilbert, in 1915, and makes use of the principle of stationary action. Given an action S, and considering a four dimensional volume V, this principle states that if we make arbitrary variations of the metric, which vanish on the boundary of V, then S must be stationary, δS/δgµν = 0. Moreover, the equations that describe our system will be the ones for which the action is stationary. He recognised two requirements for the gravitational Lagrangian. It should be a generally covariant scalar, if it is to lead to covariant equations, and it should be such that metric variation will lead to second order differential equations. This second requirement comes from the fact that there is no other theory, for gravity, which has higher-order field equations. In general, this action is given by

S = 1 2κ

Z

V

d4x√−gLgrav+ SM(gµν, Ψ ) , (2.38)

where Lgrav is the gravitational Lagrangian, SM is the matter action, Ψ denotes the matter fields and g

is the determinant of the metric tensor gµν. The action that leads to eq. (2.34) is called the E-H action

and is given by S = 1 2κ Z V d4x√−g(R + 2Λ) + SM(gµν, Ψ ) , (2.39)

where, in this case, the gravitational Lagrangian is

Lgrav,GR= R + 2Λ . (2.40)

For a universe with a cosmological constant equal to zero (Λ = 0), the gravitational Lagrangian is simply given by the Ricci scalar R, which is the simplest generally covariant scalar that one can construct that depends on the second derivatives of the metric.

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leads to the energy-momentum tensor, Tµν. Moreover, the matter action is given by SM = Z d4x√−gLM(gµν, Ψ ) , (2.41) where Tµν = δ(√−gLM) δgµν −2 √ −g . (2.42)

Let us now derive the Einstein field equations, starting with the E-H action, given by eq. (2.39). In order to do that, we have to determine its variation with respect to the metric,

δS = 1 2κ Z V d4xδ √−g(R + 2Λ) + δSM = 1 2κ Z V d4x δ√−g(R + 2Λ) +√−gδR + δSM . (2.43) We have to express the terms δ√−g(R + 2Λ) and√−gδR in terms of δgµν. Starting with the first term,

using the usual chain rule with respect to g we get δ√−g = ∂( √ −g) ∂g δg = − 1 2(−g) −1 2δg = − 1 2√−gδg . (2.44)

We can then use Jacobi’s formula, which expresses the differentiation of a determinant, in this case g, in terms of the inverse of the matrix, gµν, and of the determinant itself, yielding

δg = ∂g ∂gµν

δgµν = ggµνδgµν . (2.45)

We can also make use of the fact that δαµis a constant. Using its definition, δµα= gµνgαν, we get

δ(δαµ) = 0 ⇔ δ(gµνgαν) = 0

⇔ gµνδgαν + gανδgµν = 0

⇔ gµβgµνδgαν = −gµβgανδgµν

⇔ δβνδgαν = −gµβgανδgµν

⇔ δgαβ = −gµβgανδgµν , (2.46)

which allows us to write δgαβ in terms of δgαβ. Using eq. (2.45) and eq. (2.46), we can express δ

√ −g in terms of δgαβ, like so δ√−g = − 1 2√−ggg µνδg µν = − 1 2√−ggg µν−g ανgβµδgαβ  = − √ −g 2 gαβδg αβ . (2.47)

Using eq. (2.47), we can write eq. (2.43) as δS = 1 2κ Z V d4x  − √ −g 2 gµνδg µν(R + 2Λ) +−gδR  + δSM . (2.48)

Now, we turn to the second term in eq. (2.48), √−gδR. Recalling the definition of the Ricci scalar, given by eq. (2.32), its variation with respect to the metric is equal to

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As such, eq. (2.48) becomes δS = 1 2κ Z V d4x  −g  −1 2gµν(R + 2Λ) + Rµν  δgµν+√−ggµνδRµν  + δSM (2.50)

The first term of eq. (2.49) is already written in terms of δgµν and so, we only need to work on δRµν.

From the definition of the Ricci tensor, which is given by eq. (2.31), we understand that, in order to determine δRµν, we first need to determine the variation of the Riemann tensor, which is given by eq.

(2.16). As such, the variation of the Riemann tensor is given by

δ Rρµσν = δ ∂σΓµνρ  − δ ∂νΓµσρ  + δ Γµνα Γασρ + Γµναδ (Γασρ ) − δ Γµσα  Γανρ − Γµσα δ (Γανρ ) .

(2.51) It turns out that there is a resemblance between the previous expression and the difference between two covariant derivatives of the variation of the metric connection. The latter is given by

σδΓµνρ − ∇νδΓµσρ = ∂σδΓµνρ − ∂νδΓµσρ + δΓµναΓσαρ − δΓµσα Γναρ − δΓανρ Γσµα + δΓασρ Γνµα , (2.52)

where we have applied the covariant derivative formula, given by eq. (2.9), to the metric connection as such

σδΓµνρ = ∂σδΓµνρ + δΓµναΓσαρ − δΓµαρ Γσνα − δΓανρ Γσµα . (2.53)

Noticing that δ(∂ρΓνσµ) = ∂ρ(δΓνσµ), we find that

δ(Rρµσν) =∇σδΓµνρ − ∇νδΓµσρ , (2.54)

δ(Rµν) =δ(Rρµρν) = ∇ρδΓµνρ − ∇νδΓµρρ (2.55)

and, as a consequence, eq. (2.49) becames

δR = Rµνδgµν+ gµν ∇ρδΓµνρ − ∇νδΓµρρ



. (2.56)

Focusing on the second term of the previous expression, in the context of eq. (2.50) we have that 1 2κ Z V d4x√−ggµνδRµν = 1 2κ Z V d4x√−ggµν ∇ρδΓµνρ − ∇νδΓµρρ  . (2.57)

Using a property of the metric tensor, given by eq. (2.28), and the fact that we can use the metric to raise tensorial indices, the product gµνδRµνresults in the following covariant derivative

gµνδ(Rµν) = gµν∇ρδΓµνρ − gµν∇νδΓµρρ

= ∇ρ(gµνδΓµνρ ) − ∇µδΓµρρ

= ∇ρ(gµνδΓµνρ − gµρδΓµσσ ) = ∇ρXρ , (2.58)

where we have defined Xρ= gµνδΓµνρ − gµρδΓµσσ . Therefore, eq. (2.57) becomes

1 2κ Z V d4x√−ggµν ∇ρδΓµνρ − ∇νδΓµρρ  = 1 2κ Z V d4x√−g∇ρXρ (2.59)

This integral is called a divergence and can be written as a boundary term. Formally, we can add an extra boundary term, called Gibbons-York-Hawking boundary term, on the action, such that its variation

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with respect to the metric tensor exactly cancels the boundary contribution of the E-H term given by eq. (2.59) [28]. Without discussing in detail this procedure, we state that

Z

V

d4x√−g∇µXµ= 0 , (2.60)

and we refer to this integral as a pure divergence. Thus, using eq. (2.42), eq. (2.58) and eq. (2.60), in eq. (2.50), we get δS = 1 2κ Z V d4x  −g  −1 2gµν(R + 2Λ) + Rµν  δgµν+√−ggµνδR µν  + δSM = 1 2κ Z V d4x  − √ −g 2 gµνδg µν  (R + 2Λ) +√−g (Rµνδgµν)  + δSM = 1 2κ Z V d4x√−g  −1 2gµν(R + 2Λ) + Rµν  +δ( √ −gLM) δgµν 2κ √ −g  δgµν = 1 2κ Z V d4x√−g  −1 2gµνR − Λgµν+ Rµν− κTµν  δgµν . (2.61)

As we stated previously, the equations of motion are obtained when the action is stationary, δS = 0, for arbritary variations of the metric tensor. Therefore, since δgµν must be arbritary, the variation of the action is zero when

−1

2gµνR − Λgµν+ Rµν− κTµν = 0 ⇔ Rµν− 1

2gµνR − Λgµν = κTµν ⇔ Gµν− Λgµν = κTµν , (2.62) thus revealing the Einstein field eq. (2.34).

2.1.3 Friedmann and Raychaudhuri equations

The discovery that the universe is expanding was one of the greatest intellectual revolutions of the XX century. The cosmological constant, added by Einstein, represented a force which was embedded in space-time itself, counteracting gravity and being responsible for avoiding the collapse of the universe, allowing for a static one. During 1922, while Einstein and other physicists were trying to avoid the possibility of a non-static universe, Alexander Friedmann was willing to try to explain why this could be the case, many years even before Edwin Hubble’s discovery of an expanding universe, in 1929 [9].

The equations of GR, eq. (2.34), that determine how the universe evolves with time, turn out to be rather complicated to be solved in an exact way. As such, Friedmann made two simple, yet reasonable, assumptions: the universe is isotropic on large scales and this is true for any point in the universe, thus also being homogeneous. With such assumptions, the line element, in spherical coordinates, takes the general form ds2 = −dt2+ a(t)2  dr2 1 − kr2 + r 22+ r2sin2θdφ2  , (2.63)

where a(t) is the scale factor and k is the curvature of the universe, which can be set equal to −1, 0 or +1, depending if one is considering a hyperspherical, spatially flat or hyperbolic universe, respectively. This metric, known as the FLRW metric, is expressed with a {−1, +1, +1, +1} signature, which we will be using throughout this work. Assuming a perfect fluid description for the content of the universe, which is compatible with the scenario of an isotropic and homogeneous universe, the energy-momentum

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tensor is given by

Tµν = (ρ + P )UµUν+ P gµν , (2.64)

where

U = (U0, U1, U2, U3) = γ(1, ~u) (2.65) is the four-velocity field of an observer comoving with the fluid, ~u is the three-dimensional velocity, γ = 1/√1 − u2 is the Lorentz factor and ρ = ρ(t) and P = P (t) are the fluid’s energy density and

pressure, respectively. The magnitude of the four-velocity, UνUν = gµνUνUµ, is always equal to ±1,

depending on the choice of metric signature. With a {−1, +1, +1, +1} signature,

UνUν = −1 . (2.66)

Considering the FLRW metric, we can determine the time-time component of eq. (2.34), which is called the Friedmann equation, or the first Friedmann equation. Focusing on the time-time component of the right-hand side of eq. (2.34), we need to determine T00. In a comoving frame ~u = ~0 and, therefore,

the Lorentz factor is equal to 1. As a consequence U0U0 = 1 and, given that g00= −1,

T00= (ρ + P )U0U0+ P g00= ρ . (2.67)

Focusing on the time-time component of the left-hand side of eq. (2.34), we can determine R00and

R, using eq. (2.31) and eq. (2.32), respectively, obtaining R00= −3 ¨ a a , (2.68) R = 6(k + ˙a 2+ a¨a) a2 , (2.69)

finding the expression for G00to be equal to

G00= R00−

1

2g00R = 3H

2+3k

a2 , (2.70)

where H = ˙a/a is the Hubble parameter and an overdot denotes differentiation with respect to the time coordinate. Using eq. (2.67), eq. (2.70) and g00 = −1, we can finally compute G00− Λg00 = κT00,

resulting in H2 = κρ 3 − k a2 + Λ 3 . (2.71)

This is one of the equations for the evolution of the scale factor, a = a(t), which describes the evolution of the size of any length scale in the universe. For a spatially flat universe, k = 0, eq. (2.71) implies that the universe will continue to expand as long as there is matter in it (as long as ρ 6= 0). Additionally, there is a second equation of motion, which we will be deriving in the following discussion. If we compute the trace, defined by eq. (2.22), of eq. (2.34), we obtain an expression for the Ricci scalar

gµν(Gµν− Λgµν) = κgµνTµν ⇒ R = −κT − 4Λ . (2.72)

Multiplying this expression by −gµν/2, and subtracting eq. (2.34), allows us to obtain

Rµν = κ  Tµν− 1 2T gµν  − Λgµν . (2.73)

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The time-time component of this equation is called the second Friedmann equation or Raychaudhuri equation. At this point, we only need to compute the trace of the energy-momentum tensor, Tµν, which

gives

T =gµνTµν =

=(ρ + P )gµνUµUν+ P gµνgµν =

=(ρ + P )UνUν + 4P =

=3P − ρ . (2.74)

Using this result, eq. (2.68) and eq. (2.69), the time-time component of eq. (2.73) becomes ¨ a a = − κ 6 (ρ + 3P ) + Λ 3 . (2.75)

While eq. (2.71) is an equation in ˙a, describing the velocity of the expansion/contraction, eq. (2.75) involves ¨a, which therefore is related to the acceleration of the expansion/contraction. Contrary to the velocity, the acceleration does not depend on the spatial curvature of the universe, since k does not appear in eq. (2.75). The minus sign on the first two terms implies that the expansion will always be slowed by gravity. The last term, the one containing the cosmological constant, is positive and it is, therefore, the one responsible for an accelerated expansion.

It is often assumed that during several periods of the history of the universe, the energy density and pressure are related with each other, through an EoS with a simple form

P = wρ, −1 ≤ w ≤ 1 , (2.76)

where w is a constant. In this context, for non-relativistic matter w = 0, for radiation w = 1/3, a cosmological constant has w = −1, a free scalar field has w = 1 and curvature has w = −1/3.

Forming a consistent set of equations, together with eq. (2.71) and eq. (2.75), the energy-momentum conservation law, meaning the conservation of the energy-momentum tensor, ∇νTµν = 0, yields

˙

ρ + 3H(ρ + P ) = 0 . (2.77)

2.2

f (R) gravity

There are many ways we can change the gravitational Lagrangian of GR in order to build a theory that includes higher-order curvature invariants. The one we will be considering in this work is the class of f (R) theories in which, put simply, we replace the Ricci scalar R in eq. (2.40) by a function of it, f (R). Having said that, there are three different ways we can proceed with the variation of the action or, in other words, there are three different formalisms, namely, metric formalism, Palatini formalism and metric-affine formalism.

In the standard metric variation, or metric formalism, the metric tensor and the connection are as-sumed to be dependent variables, i.e. the affine connection is the metric connection, given by eq. (2.27), which is the case in GR. In the Palatini formalism we take a step forward and consider the opposite case in which the metric and the connection are independent variables. In that case, we have to compute the variation of the action with respect to both of them separately, under the assumption that the matter ac-tion does not depend on the connecac-tion. Both of these formalisms, metric and Palatini, lead to the same

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field equations for an action whose Lagrangian is linear in R. That is no longer true for a more general action, with a Lagrangian with second or higher-order terms. Lastly, we can also consider the metric-affine formalism, in which the metric and the connection are assumed to be independent variables, as in the Palatini formalism, but the matter action depends on the connection. In the last two cases, since it is assumed that the affine connection is not the metric connection, this means that the torsion tensor does not necessarily vanish. At this point, it is worth mentioning that it is well known that torsion, also being part of the geometric description of a generic theory of gravity, can represent the geometric counterpart of spin as mass/energy is the source of curvature [8].

2.2.1 Lagrangian formulation of metric f (R) gravity

In this work, we will be considering metric f (R) gravity, for which we will derive its corresponding field equations, using the principle of stationary action. The gravitational Lagrangian in the framework of GR is given by eq. (2.40). In the context of f (R) gravity, the Lagrangian density is a function of the Ricci scalar, Lgrav = f (R). Thus, the corresponding action is given by

S = 1 2κ

Z

V

d4x√−gf (R) + SM(gµν, Ψ ) . (2.78)

The steps to obtain the field equations, in the framework of metric f (R) gravity, will now be pre-sented in quite some detail, although the majority of steps was already prepre-sented when deriving the Einstein field equations. Once again, we want to determine the variation of the action, in this case eq. (2.78), with respect to the metric, as such

δS = 1 2κ

Z

V

d4xδ √−g f (R) +√−gδf (R) + δSM(gµν, Ψ ) . (2.79)

Regarding the first term, we have already computed δ√−g, in eq. (2.47), when deriving the Einstein field equations. Concerning the second term, we can use the chain rule to writte δf (R) = f0(R)δR, where the prime in f0(R) indicates differentiation with respect to R. In that sense, the action becomes

δS = 1 2κ Z V d4x  − √ −g 2 gµνδg µνf (R) +−gf0 (R)δR  + δSM(gµν, Ψ ) . (2.80)

Using eq. (2.49), we get δS = 1 2κ Z V d4x  −g  −1 2gµνf (R) + f 0 (R)Rµν  δgµν+√−gf0(R)gµνδRµν  + δSM . (2.81)

Contrary to the situation we have when deriving the Einstein field equations, where we have the term √

−ggµνδR

µν in eq. (2.50), in this case we have

−gf0(R)gµνδR

µν. Because of the extra f0(R), we

do not immediately have a pure divergence, as in eq. (2.60) and, therefore, we need to proceed with our calculations in a different way. We have already seen that the variation of the Ricci tensor, δRµν, is given

by eq. (2.55). Using

δΓβασ = 1 2g

σγ[∇

βδgαγ+ ∇αδgβγ − ∇γδgβα] , (2.82)

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second and third terms of (2.56), obtaining δRµν =∇ρδΓµνρ − ∇νδΓµρρ =∇ρ  1 2g ργ(∇ µδgνγ+ ∇νδgµγ− ∇γδgµν)  − ∇ν 1 2g ργ(∇ µδgργ+ ∇ρδgµγ− ∇γδgµρ)  =1 2g ρσ  ∇ρ∇µδgνσ− ∇ρ∇σδgµν− ∇ν∇µδgρσ+ ∇ν∇σδgµρ+ [∇ρ, ∇ν]δgµσ  . (2.83) Using eq. (2.18), on the last two terms of the previous expression, we get

δRµν = 1 2g ρσ  ∇ρµδgνσ− ∇ρ∇σδgµν− ∇ν∇µδgρσ+ (−Rαµνσδgαρ− Rαρνσδgµα + ∇σ∇νδgµρ) + (−Rαµρνδgασ− Rασρνδgµα)  = 1 2  gρσ(∇ρ∇µδgνσ+ ∇σ∇νδgµρ) − gρσRαµρνδgασ− gρσRασρνδgµα −2δgµν − gρσ∇νµδgρσ− gρσRαµνσδgαρ− gρσRαρνσδgµα  , (2.84)

where2 = ∇σ∇σ is notation for the d’Alembertian operator. Using eq. (2.46), we can show that

− gρσ∇ν∇µδgρσ = gρσ∇ν∇µδgρσ . (2.85)

We can use the properties of the Riemann tensor to also show that

gρσRαµρνδgασ = Rα σµ νδgασ = −Rαµνσδgασ = −gρσRαµνρδgασ = −gρσRαµνσδgαρ (2.86)

and

gρσRαρνσδgµα= Rαρνρδgµα= −Rα ρρ νδgµα = −gβρRαρβνδgµα= −gρσRασρνδgµα . (2.87)

As such, using eq. (2.85), eq. (2.86) and eq. (2.87), as well applying the metric tensor on the first two terms of eq. (2.84), this equation becomes

δRµν = 1 2  ∇σ∇µδgνσ+ ∇ρ∇νδgµρ−2δgµν+ gρσ∇ν∇µδgρσ  . (2.88)

Finally, we use eq. (2.46), to write all the terms in eq. (2.88) in terms of δgαβ, getting δRµν = 1 2  − ∇σ∇µgασgνβδgαβ− ∇ρ∇νgαρgµβδgαβ +2gανgβµδgαβ+ gρσ∇ν∇µδgρσ  =1 2 h −∇α∇µgβνδgαβ− ∇α∇νgβµδgαβ +2gανgβµδgαβ+ gαβ∇ν∇µδgαβ i . (2.89) Therefore, gµνδRµν = 1 2h2δ ν βgανδgαβ+ gαβ∇ν∇νδgαβ− δβµ∇α∇µδgαβ− δνβ∇α∇νδgαβ i = 1 2h2gαβδg αβ+ g αβ2δgαβ − ∇α∇βδgαβ− ∇α∇βδgαβ i = gαβ2δgαβ − ∇α∇βδgαβ (2.90)

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Substituting this result in eq. (2.81), we get δS = 1 2κ Z V d4x√−gh−gµν 2 δg µνf (R) + f0 (R) (Rµνδgµν+ gµν2δgµν− ∇µ∇νδgµν) i + δSM . (2.91) There are still two things we can do to further simplify eq. (2.91). In fact, we can only use the principle of least action when δgµν is isolated in the expression that is being integrated. For now, let us consider the 3rd and 4th terms of eq. (2.91). Focusing on the 3rd one, we can integrate by parts, twice, getting √ −gf0(R)gµν2δgµν = √ −gf0(R)gµν∇ρ∇ρδgµν =√−g[∇ρ f0(R)gµν∇ρδgµν − ∇ρ f0(R)gµν ∇ρδgµν] = −√−g∇ρ f0(R)gµν ∇ρδgµν = −√−g[∇ρ(∇ρf0(R)gµνδgµν) − ∇ρ∇ρ(f0(R)gµν)δgµν] =√−g2f0(R)gµνδgµν , (2.92)

where the integration is implicit in the calculations. The vanishing of ∇ρ(f0(R)gµν∇ρδgµν), in the

third line of eq. (2.92), and ∇ρ(∇ρf0(R)gµνδgµν), in the forth, is due to the fact that they are pure

divergences, as the corresponding integrals are of the form given in eq. (2.60). We can do exactly the same procedure with the 4th term of (2.91), leading to

√ −gf0(R)∇µ∇ν(δgµν) = √ −g[∇µ(f0(R)∇νδgµν) − ∇µ(f0(R))∇νδgµν] = −√−g∇µ(f0(R))∇νδgµν = −√−g[∇ν(∇µ(f0(R))δgµν) − ∇µ∇ν(f0(R))δgµν] =√−g∇µ∇ν(f0(R))δgµν . (2.93)

Substituting eq. (2.92), eq. (2.93) and eq. (2.42), in eq. (2.91), while also factorizing δgµν, eq. (2.91) becomes δS = 1 2κ Z d4x√−g  −1 2gµνf (R) + f 0(R)R µν+2(f0(R))gµν− ∇µ∇ν(f0(R)) − κTµν  δgµν . (2.94) Through the principle of least action, given that δgµν must be arbitrary, the field equations for f (R) gravity are the ones for which δS = 0. These are given by

−1

2gµνf (R) + f

0(R)R

µν− [∇µ∇ν− gµν2]f0(R) = κTµν (2.95)

and the corresponding trace is given by

f0(R)R − 2f (R) + 32f0(R) = κT . (2.96)

It is possible to write eq. (2.95) in the form of Einstein equations with an effective stress-energy tensor composed of curvature terms moved to the right hand side, like so

Gµν = Gef f



Tµν+ Tµν(ef f )



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where Tµν(ef f )= 1 κ  gµν [f (R) − Rf0(R)] 2 + ∇µ∇νf 0 (R) − gµν2f0(R)  (2.98) and Gef f ≡ G/f0(R) (2.99)

can be regarded as an effective gravitational coupling constant. Positivity of Gef f, equivalent to the

requirement that the graviton is not a ghost, therefore imposes that f0(R) > 0 [21]. Moreover, if f00(R) > 0 the Dolgov-Kawasaki instability is avoided. This stability condition, which expresses that the scalar degree of freedom is not a ghost, can be given a physical interpretation [29]. Assuming that f0(R) > 0 applies then, if the gravitational coupling decreases as R increases, as such

dGef f

dR =

−f00(R)G

(f0(R))2 < 0, (2.100)

a mechanism operates which compensates for the increase of R that comes from eq. (2.96), which is a dynamical equation on R that shows that R generates larger curvature, the effect of which becomes stronger with increased Gef f. This can only be achieved if f00(R) > 0.

2.2.2 Generalized Friedmann and Raychaudhuri equations

In the context of GR, assuming a FLRW metric, we have obtained eq. (2.71) and eq. (2.75). We can consider the same metric and deduce the corresponding equations in f (R) gravity, in order to see how they differ from the previous ones. Using g00= −1, eq. (2.68) and eq. (2.67), the time-time component

of eq. (2.95) is given by 1 2f (R) − 3 ¨ a af 0 (R) − [∇0∇0+2]f0(R) = κρ (2.101)

Focusing on the last terms in the left-hand side of this equation, we have that [∇0∇0+2]f0(R) = ∂0∂0f0(R) + gµν∇µ(∂νf0(R))

= ∂0∂0f0(R) + gµν∂µ∂νf0(R) − gµνΓνµη ∂ηf0(R)

= ∂0∂0f0(R) − ∂0∂0f0(R) − gµνΓνµ0 ∂0f0(R)

= −gµνΓνµ0 f˙0(R) , (2.102)

where we have used eq. (2.8), for Xν = ∂νf0(R) and the fact that the covariant derivative of the scalar

f0(R) is the usual partial derivative, ∇νf0(R) = ∂νf0(R). Notice that, in the implicit summations in the

third line of eq. (2.102), only the terms with partial derivatives with respect to time are relevant, since f0(R) only depends on time. Assuming a FLRW metric, using eq. (2.27), we can determine that

gµνΓνµ0 = 3˙a

a . (2.103)

We can express ˙f0(R) in terms of derivates with respect to R. As such, using the chain rule, we get ˙ f0(R) = d dt d dRf (R) = dR dt d dR df (R) dR = ˙Rf 00 (R) . (2.104)

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Using eq. (2.102), eq. (2.103) and eq. (2.104), eq. (2.101) becomes 1 2f (R) − 3 ¨ a af 0(R) + 3˙a a ˙ Rf00(R) = κρ (2.105)

The Ricci scalar, given by eq. (2.69), is given by

R = 6 " ¨ a a+  ˙a a 2# (2.106)

with k = 0. Using this, we can write

3¨a a = 1 2R − 3  ˙a a 2 , (2.107)

which we can substitute in eq. (2.105), getting  ˙a a 2 − 1 3f0(R)  1 2f (R) − Rf 0(R) − 3 ˙a a  ˙ Rf00(R)  = κρ 3f0(R) . (2.108)

This is the modified Friedmann equation in the context of metric f (R) gravity. We can observe that, setting f (R) = R, we have f0(R) = 1 and f00(R) = 0 and eq. (2.108) reduces to the standard Friedmann equation, given by eq. (2.71), with k = Λ = 0. It is clear that eq. (2.108) is a higher than second order equation, given the presence of ˙R and eq.(2.106), which poses a problem that we will later discuss. In a similar way, we can determine the space-space components of eq. (2.95), which give [30]

2 ¨a a  + ˙a a 2 + 1 f0(R)  2 ˙a a  ˙ Rf00(R) + ¨Rf00(R) + ˙R2f000(R) −1 2f (R) − Rf 0(R)  = − κP f0(R) . (2.109) Finally, if we introduce, in eq. (2.109), the expression we get for ( ˙a/a)2, from eq. (2.108), we obtain

 ¨a a  + 1 2f0(R)  ˙a a  ˙ Rf00(R) + ¨Rf00(R) + ˙R2f000(R) − 1 3[f (R) − Rf 0(R)]  = − κ 6f0(R)(ρ+3P ) , (2.110) which is the modified Raychaudhuri equation in the context of metric f (R) gravity. This equation is also a higher than second order differential equation. Once again, setting f (R) = R, we have f0(R) = 1 and f00(R) = f000(R) = 0 and eq. (2.110) reduces to the standard Raychaudhuri equation, given by eq. (2.75), with Λ = 0.

Figure

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