UNIVERSIDADE ESTADUAL DE CAMPINAS

Instituto de Física “Gleb Wataghin”

Ana Rubiela Romero Castellanos

Higher derivative corrections to the R +R2

inflationary model and axions in the 3− 3− 1 model

Correções de derivadas superiores ao modeloinflacionário R +R2 e áxions no modelo 3− 3− 1

CAMPINAS2019

Ana Rubiela Romero Castellanos

Higher derivative corrections to the R +R2

inflationary model and axions in the 3− 3− 1 model

Correções de derivadas superiores ao modeloinflacionário R +R2 e áxions no modelo 3− 3− 1

Thesis presented to the Institute of Physics“Gleb Wataghin” of the University of Campi-nas in partial fulfillment of the requirementsfor the degree of Doctor in Sciences.

Tese apresentada ao Instituto de Física“Gleb Wataghin” da Universidade Estadualde Campinas como parte dos requisitos exi-gidos para a obtenção do título de Doutoraem Ciências.

Orientador: Prof. Dr. Pedro Cunha de HolandaEste exemplar corresponde à versão finalda tese defendida pela aluna Ana RubielaRomero Castellanos e orientada pelo Prof.Dr. Pedro Cunha de Holanda.

CAMPINAS2019

Ficha catalográficaUniversidade Estadual de Campinas

Biblioteca do Instituto de Física Gleb WataghinLucimeire de Oliveira Silva da Rocha - CRB 8/9174

Romero Castellanos, Ana Rubiela, 1980- R664h RomHigher derivative corrections to the $R+R^2$ inflationary model and axions

in the 3-3-1 model / Ana Rubiela Romero Castellanos. – Campinas, SP : [s.n.],2019.

RomOrientador: Pedro Cunha de Holanda. RomTese (doutorado) – Universidade Estadual de Campinas, Instituto de

Física Gleb Wataghin.

Rom1. Starobinsky, Inflação de. 2. Correções de derivadas superiores. 3.

Aproximação de slow-roll. 4. Matéria escura (Astronomia). 5. Áxions. I.Holanda, Pedro Cunha de, 1973-. II. Universidade Estadual de Campinas.Instituto de Física Gleb Wataghin. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Correções de derivadas superiores ao modelo inflacionário$R+R^2$ e áxions no modelo 3-3-1Palavras-chave em inglês:Starobinsky inflationHigher-derivative correctionsSlow-roll approximationDark matter (Astronomy)AxionsÁrea de concentração: FísicaTitulação: Doutora em CiênciasBanca examinadora:Pedro Cunha de Holanda [Orientador]Orlando Luis Goulart PeresDonato Giorgio TorrieriAlex Gomes DiasCelso Chikahiro NishiData de defesa: 18-09-2019Programa de Pós-Graduação: Física

Identificação e informações acadêmicas do(a) aluno(a)- ORCID do autor: https://orcid.org/0000-0002-6115-229X- Currículo Lattes do autor: http://lattes.cnpq.br/0520074569209081

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MEMBROS DA COMISSÃO JULGADORA DA TESE DE DOUTORADO DE ANA RUBIELA ROMERO CASTELLANOS – RA 180529 APRESENTADA E APROVADA AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 18 / 09 / 2019.

COMISSÃO JULGADORA:

- Prof. Dr. Pedro Cunha de Holanda – Orientador – DRCC/IFGW/UNICAMP - Prof. Dr. Orlando Luis Goulart Peres – DRCC/IFGW/UNICAMP - Prof. Dr. Donato Giorgio Torrieri – DRCC/IFGW/UNICAMP - Prof. Dr. Alex Gomes Dias – CCNH/UFABC - Prof. Dr. Celso Chikahiro Nishi – CMCC/UFABC

OBS.: Ata da defesa com as respectivas assinaturas dos membros encontra-se no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria do Programa da Unidade.

CAMPINAS 2019

Acknowledgements

First, I am highly grateful with my supervisor, professor Pedro Cunha de Holanda,who has given me great support and provided the opportunity to work in an au-tonomous way during this Ph.D., but whenever help was needed he was ready toaiding and heartening.

Through my Ph.D. I had the opportunity to collaborate with excellent academics.I want to especially thank professor Flavia Sobreira from UNICAMP, who trustedme, taught me that surrendering is not an option and set up the opportunity to workwith her and with great physicists, as professor Ilya Shapiro from UFJF, who duringmy stay in UFJF was a great tutor and advisor, and taught me a lot of things. Ihave to say that it was a huge privilege to have the opportunity to work with oneof the creators of the inflationary theory, professor Alexei A. Starobinsky, whom Iwant to thank for his guidance and all his comments and suggestions during severalstages of this work.

Also, I desire to express my gratitude to professors Juan Carlos Montero (IFT)and Bruce Lehmann Sánchez Vega (UFABC), for an efficient and productive collab-oration.

I want to thank in a very special way to the MSc. Carlos Enrique Alvarez Salazar,not just by his work in constructive academic collaborations but, more important,to make more pleasant the long days of work with his peculiar addiction to Mozart,to transform the bad coffees in pleasant moments and because due to him I neverfelt alone.

Finally, I want to thank the “Gleb Wataghin” Institute of Physics at UNICAMP,for the opportunity to curse my Ph.D. studies.

This study was financed in part by the Coordenação de Aperfeiçoamento dePessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Resumo

Nesta tese, analisaram-se dois tópicos: o modelo inflacionário de Starobinsky comum termo de derivada superior como uma pequena perturbação, e a inclusão deáxions num modelo SU(3)c⊗SU(3)L⊗U(1)X (3−3−1) como um possível candidatopara matéria escura.

Na primeira parte dessa tese, o termo RR é adicionado no modelo de Staro-binsky como uma pequena perturbação, e é mostrado que é possível ter uma épocainflacionária no universo primitivo e, ao mesmo tempo, dar valores para o índice es-pectral e a razão escalar-tensor que são consistentes com os vínculos observacionaisatuais sob essas quantidades.

Na segunda parte desse trabalho, foi obtida a abundância de matéria escurana forma de áxions no contexto de um modelo 3 − 3 − 1, levando em conta suascontribuições do mecanismo de desalinhamento, e devido ao decaimento de defeitostopológicos, domínios de parede e sistemas corda-parede. Os resultados foram com-parados com observações atuais, e encontrou-se o espaço de parâmetros permitido,mostrando que o áxion neste cenário pode ser um candidato interessante à matériaescura. Finalmente, foi feita uma comparação de três versões diferentes de modelo3 − 3 − 1 com áxions em seu espectro físico, mostrando as diferenças, vantagens edesvantagens de cada modelo, quando os vínculos atuais sob a abundância relíquiade matéria escura e perspectivas de detecção de áxions são levadas em conta.

Keywords: Inflação de Starobinsky, Correções de derivadas superiores, Aproxi-mação de slow-roll, Matéria escura, Áxions, Modelo 3-3-1.

Abstract

In this thesis, we address two topics: the Starobinsky inflationary model with ahigher derivative term as a small perturbation, and the inclusion of axions in aSU(3)c ⊗ SU(3)L ⊗ U(1)x (3− 3− 1) model as a possible dark matter candidate.

In the first part of the thesis, the term RR is added to the Starobinsky modelas a small perturbation, and it is shown that it is possible to have an inflationaryepoch in the early universe and, at the same time, give values for the spectral indexand the scalar-to-tensor ratio which are consistent with the current observationalconstraints on this quantities.

In the second part of this work, we obtained the abundance of axion dark matterin the context of a 3 − 3 − 1 model, taking into account its contributions from themisalignment mechanism, and due to the decay of topological defects, domain wallsand string-walls systems. The results were compared with current observations, andwe found the allowed parameter space, showing that the axion in this scenario canbe an appealing candidate to dark matter. Finally, we made a comparison of threedifferent versions of 3− 3− 1 model with axions in their physical spectrum, display-ing the differences, pros, and cons for each model when the present constraints onthe relic abundance of dark matter and prospects of axion detection are taken intoaccount.

Keywords: Starobinsky inflation, higher-derivative corrections, slow-roll approxi-mation, dark matter, axions, 3− 3− 1 model.

List of Figures

2.1 The energy scale of the new scalaron χ as a function of the perturba-tive parameter k. The shaded regions are excluded by the constraintsimposed by Planck data [13], as will be shown later. . . . . . . . . . . 33

2.2 RR correction to the Starobinsky model of inflation for differentvalues of the parameter k. The black continuous line corresponds tothe original R+ αR2 model, and the colored dashed lines to nonzerovalues of k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 The slow roll parameters ε (up) and η (down) for the model withRR term as functions of the inflaton field χ. . . . . . . . . . . . . . 36

2.4 Observational constraints set by the Planck collaboration [13, 71] onthe scalar-to-tensor ratio r and scalar spectral index ns, and the pre-diction for these quantities in the R+R2 +RR model. The allowedregions for this model are presented in red and green color for nega-tive and positive values of the k parameter, respectively. This figureis an original result of this thesis and appears in [12]. . . . . . . . . . 38

2.5 Observational constraints on the slow roll parameters set by Planckcollaboration [89], and the prediction for these quantities in the R +αR2+γRR. The subindex V indicates that the slow roll parametersare written as a function of the scalar potential. This figure is anoriginal result of this thesis and appears in [12]. . . . . . . . . . . . . 39

3.1 Relic density of non-thermal axion dark matter in the 3−3−1 model,assuming exact scaling, p = 1, and |g| = 1. The central values of theparameters in Eqs. (3.35) and (3.39) together with NDW = 3 havebeen used. The vertical dashed lines limit regions with over pro-duction of axions by decay of domain walls (left line) and strings(right line), while the horizontal red line is the experimental con-straint Ωah

2 = ΩPlanckDM h2, the cyan and black lines show the axion

abundances produced by misalignment and global string decay mech-anisms, respectively. Finally, the blue lines show the abundance ofaxion dark matter due to the decay of domain wall systems forN = 10and N = 11, calculated for the coupling constant value |g| = 1. . . . . 53

3.2 Observational constraints on the parameter space fa − |g| in the 3−3−1 model, assuming exact scaling (a) and deviation from scaling (b).These plots correspond to the Z10 discrete symmetry, and NDW = 3.The shaded regions in light red and light blue correspond to regionsof the parameter space where the constraints given by ma,QCD >ma,gravity and Ωah

2 6 ΩPlanckDM h2 are violated, respectively. Moreover,

the regions above the straight red lines correspond to excluded regionsset by the NEDM condition, as given by Eqs. (3.21) and (3.22), forthree different choices of the δD parameter. . . . . . . . . . . . . . . 55

3.3 Projected sensitivities of different experiments in the search for axiondark matter. The green regions show sensitivities of light-shining-through-wall experiments like ALPS-II [103], of the helioscope IAXO [102],of the haloscopes ADMX and ADMX-HF [104, 105]. The yellow bandcorresponds to the generic prediction for axion models in QCD. Inaddition, the two (one) thick red (blue) lines stand for the predictedmass ranges and coupling to photons in this model, for |g| = 0.1(|g| = 1), where axions make up the total DM relic density. . . . . . . 56

3.4 Axion abundance in the 3−3−1 models considered in this work, char-acterized by different orders of the discrete symmetry stabilizing theaxion solution to the strong CP problem and numbers of domain wallsNDW. The blue bands in each panel give the total axion abundance,which can be compared with the dark matter abundance measured by[13], given by the horizontal red line. The shadowed region in yellowcorresponds to values of the axion decay constant fa for which thegravitational mass induced by high dimensional operators is greaterthan the QCD axion mass. . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Constraints on the coupling g of gravitationally induced operatorsand the axion decay constant fa, for the models with NDW = 2 (up)and NDW = 3 (down). The blue region corresponds to points wherethe axion abundance is greater than the DM abundance reported by[13]. The yellow region is excluded due to the constraint ma, QCD >ma, grav. The green lines correspond to the constraint given by Eq.(3.22), where the allowed parameter space is below the lines. . . . . . 63

3.6 Sensitivity to the axion-photon coupling gaγγ for different experi-ments, adapted from [156]. The small colored lines show the ranges ofthe axion mass for which the total DM of the universe can be made ofaxions, where the numbers in brackets alongside each line representthe number of domain walls in the model, NDW, and the assumedvalue of the |g| coupling. [15] . . . . . . . . . . . . . . . . . . . . . . 64

List of Tables

3.1 The U(1) symmetry charges in the Lagrangian given by Eqs. (3.7)and (3.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 The U(1)PQ charges in the model with a Z2 discrete symmetry suchthat χ→ −χ, u4R → −u4R, and d(4,5)R → −d(4,5)R . . . . . . . . . . . 46

3.3 The charge assignment for ZD that stabilizes the PQ mechanism inthe considered 3− 3− 1 model. . . . . . . . . . . . . . . . . . . . . . 47

3.4 UPQ charge of the color-charged particles.Table adapted of [155]. . . . 58

Contents

Introduction 13

I R +R2 inflationary model 16

1 Inflationary cosmological standard model 171.1 The standard big-bang cosmology and motivations for the inflationary

paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2 Inflationary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.1 Slow-roll approximation . . . . . . . . . . . . . . . . . . . . . 211.2.2 Cosmological perturbations in inflation . . . . . . . . . . . . . 22

2 On higher derivative corrections to the R +R2 inflationary model 242.1 The R +R2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Treating RR term as a new degree of freedom . . . . . . . . . . . . 26

2.2.1 Weak field approximation . . . . . . . . . . . . . . . . . . . . 282.2.2 Simplifying Ansatz R = r1R + r2 . . . . . . . . . . . . . . . 29

2.3 Treating RR term as a small perturbation to the R +R2 model . . 302.3.1 Slow-roll conditions . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

II Axions in the 3− 3− 1 model 41

3 Axion dark matter in the 3− 3− 1 model 423.1 A brief review of the model . . . . . . . . . . . . . . . . . . . . . . . 433.2 Implementing a gravity stable PQ mechanism in the model . . . . . . 453.3 The Production of Axion Dark Matter . . . . . . . . . . . . . . . . . 48

3.3.1 Misalignment mechanism . . . . . . . . . . . . . . . . . . . . . 483.3.2 Decay of global strings . . . . . . . . . . . . . . . . . . . . . . 503.3.3 Decay of string-wall systems . . . . . . . . . . . . . . . . . . . 50

3.4 Observational constraints on the production of axion dark matter . . 523.5 Models with different discrete symmetries and number of domain walls 573.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Conclusions 65

Bibliography 66

13

Introduction

Current cosmological observations reveal that the universe is expanding and cooling,which allows us to intuit that in very early epochs there were extreme conditions ofdensity and temperature.

But how the universe could evolve from such conditions to be what we observetoday, that is, planets, stars, galaxies, clusters of galaxies, etc? This is a questionthat can not yet be answered conclusively.

The observable universe can be described by the standard cosmology (SC), amodel that has seven parameters [1]. Five of them determine the background ofthe universe: H0, ΩB, ΩDM , ΩΛ, ΩR, where H0 is the Hubble parameter measuredtoday, and Ωi = ρi/ρc with i = B, DM, Λ, R, representing the abundances ofbaryons, dark matter, dark energy and radiation, respectively, calculated as theratio of the density of the corresponding component (ρi) and ρc, the energy densitythat makes the universe to be flat, called the critical density. The two remainingparameters, the amplitude As and the spectral index ns characterize the spectrumof the initial fluctuations, the seeds for the structures that we see today in theuniverse, and that should be printed on the cosmic microwave background (CMB).Nevertheless, anything can be said about the origin of these fluctuations inside theSC.

The inflationary paradigm claims that around ∼ 10−40 to 10−32 seconds afterthe big-bang, the universe developed an inflationary era in which the scale factor ofthe universe increased in an accelerated way. It initially emerged as a mechanismto answer some questions like why the universe is so flat or why the universe ishomogeneous, among others, but soon after its proposal it was realized that in thiskind of scenarios the origin of primordial fluctuations can be explained and is relatedwith the quantum perturbations of the field(s) that drive the inflationary process.

Currently, there are a lot of models describing an inflationary phase in the earlyuniverse, which can be discriminated by their predictions for the spectral index[2]. The simplest model uses a scalar field, called inflaton, with a scalar potentialsufficiently flat in order to disregard the kinetic energy of the field and assume thatthe energy density is dominated by the potential, what constitutes the so-called slow-roll conditions, and in this case, the cosmological consequences are well understood.

Another possibility is to consider multiple scalar fields driving inflation, but itsanalysis is more involved [3]. It is important to remark here that whatever thenumber of fields in the models, they always require the addition of new fields in thetheory, not included in the standard model of particle physics (SM).

Another possibility, which avoids the ad hoc introduction of new fields, consists

14

in the analysis of corrections to the gravitational action, including terms involvinghigher powers of the curvature tensor, which is a natural way to obtain inflationaryscenarios, without the previous imposition of the existence of the inflaton field orany number of new fields [4].

The model R + R2, first proposed by Starobinsky [5], is the simplest extensionof the Einstein-Hilbert Lagrangian, and has the characteristic that the equations ofmotion include fourth-order derivatives of the metric instead of only second-orderones [6], and provided an inflationary solution even when the cosmological constantΛ is zero [7]. One feature that makes this kind of model interesting is the fact thatit can be mapped in a scalar inflationary model through a conformal transformation[8], making the comparison with observations straightforward.

In Refs. [4, 9] it was shown that when the term RR is added to the Starobinskymodel, sixth-order derivatives of the metric appear in the equations of motion, andan inflationary era in the universe can be developed. This model is conformallyequivalent to general relativity with two minimally coupled scalar fields, that is,there are two additional degrees of freedom in the model.

The main topic of the first part of this thesis, is to present a new way to treatthe extra RR term as a small perturbation to the Starobinsky model, and showthat this theory is consistent with observations and also with some previous resultsobtained when the theory is mapped to a model with two scalar fields [4, 9, 10], andwith recent results when a scalar and a vector field are used [11].

The results of this part appear in the paper [12], published in collaboration withFlavia Sobreira, Ilya L. Shapiro and Alexei A. Starobinsky. This work opens thepossibility to treat other higher derivative terms as perturbations of the Starobinskymodel and not of the Einstein-Hilbert action during the inflationary era, which isquite natural if we observe that for this phase the R2 term plays an important role.

Going back to the discussion about the parameters of the SC, we were alsointerested in the abundance of dark matter, measured in terms of the cosmologicalparameter ΩDM . The last reports of the Planck collaboration [13] show that around95% of our universe is composed of something about which we do not know anything.These results point to the conclusion that 24 and 71% of the energy of the universeare in the form of dark matter and dark energy, respectively, and just 5% is madeof baryonic matter, that can be described by the SM.

In order to describe this unknown matter in the universe, numerous candidateshave been proposed: weakly interacting massive particles (WIMPs), axions, Kaluza-Klein particles, massive astrophysical compact halo objects (MACHOs),..., but anyattempt to use a new particle to explain the features of DM needs models beyond theSM. Within the large number of candidates for DM, axions look like a compellingone due to the fact that they can solve the strong CP problem and also explainthe abundance of dark matter in the universe, evading the current experimentalconstraints on these particles.

In the second part of this thesis, we work in the context of models with a largersymmetry group than the one of the SM, specifically with the model based onSU(3)c ⊗ SU(3)L ⊗ U(1)x gauge group, called the 3 − 3 − 1 model, in order toinclude an axion in its physical spectrum, analyze the different mechanisms for its

15

production during the early universe, giving a relic abundance consistent with thecurrent constraints, and determine its prospects of detection.

The 3− 3− 1 models are appealing extensions of the SM, because these kind ofmodels have several interesting characteristics. For example, the number of fermiongenerations can be explained, also heavy top quark mass is natural in these models,and in some of the versions there are candidates to WIMPs, that can explain thetotal abundance of dark matter in the universe. On the other hand, in these scenariosneutrinos are massive particles and, it has been shown that new scalars in the modelcan contribute to the muon anomalous magnetic moment.

We analyze the scenario where the Peccei-Quinn (PQ) symmetry is broken afterthe inflationary era. In such a case, in addition to the misalignment mechanism forthe production of axions, the decay of topological defects (domain walls and string-walls systems) also contributes to the abundance of axion dark matter. By firsttime we obtain the total abundance for axions in the case of a 3− 3− 1 model andcompare it with current observations. The results of these analyses appear in thepaper [14], published in collaboration with J. C. Montero and B. L. Sánchez-Vega.

One interesting result of this study is that the order N of the discrete symmetryused to stabilize the axion mass against gravitational effects is strongly constrainedby observations on the DM relic abundance, and depending on the characteristicsof the model it can not be greater than N = 10.

Due to the fact that there are several versions of the 3− 3− 1 model includingaxions in their physical spectrum, differentiated, for example, by their particle con-tents, the order of stabilizing symmetry, PQ charges and number of domain walls,a study of the observational constraints on the parameter space for three differentversions of the 3 − 3 − 1 model with axions was made, analyzing how the numberof domain walls and the particle content of the models can modify the axion DMabundance of each model. Finally, we analyze both the QCD and gravitationalcontribution to the axion mass, and not just the linear approximation, as done inprevious works involving axions in other models. The main results of these analyseswere published in [15], written in collaboration with C. E. Alvarez-Salazar and B.L. Sánchez-Vega.

Finally, the main conclusions of this work are drawn.

Part I

R +R2 inflationary model

16

17

Chapter 1

Inflationary cosmological standardmodel

Current observations point to the establishment of the Standard Cosmological Model(SC) as a suitable description of the universe on large scales. The foundations ofthe SC are surprisingly simple: the universe is homogeneous in sufficiently largescales, and it has evolved in a nearly homogeneous way since the cosmic microwavebackground radiation (CMB) was emitted.

Nevertheless, SC possesses some issues which will be described later, as flatness,horizon and unwanted relic problems. A solution to these problems is found ininflationary theories, introduced in the early ’80s [5, 16, 17, 18] as a mechanism tosolve some of the issues of the SC.

Currently, the inflationary paradigm is one of the most accepted to explain theobserved anisotropies in the CMB [19], which were first observed by Penzias andWilson [20].

There are hundreds of inflation models, in this work we will study a model thatstands out for its simplicity and the great agreement with observations, it is theStarobinsky model R + R2 [5], which, as will be seen in the next chapter, can bemapped in one of the best known and also the simplest model, inflation driven by ascalar field.

In this chapter, we summarize the fundamental characteristics of the standardcosmological model in the first section, where also the main issues of SC are describedalong with a few words about how an inflationary era can solve such problems. After,in section two, we present the inflation driven by a scalar field, assuming slow-rollconditions. Finally, the principal results for the cosmological perturbation wheninflation is steered by a scalar field are presented.

1.1 The standard big-bang cosmology and motiva-tions for the inflationary paradigm

Our understanding of the universe is based on the cosmological model of Friedman-Robertson-Walker (FRW) [21]. This model is grounded on the cosmological princi-

18

ple, which asserts that on sufficiently large scales the universe becomes homogeneousand isotropic [22].

In this case, the spacetime metric can be written as [3, 23]

ds2 = gµνdxµdxν = −dt2 + a2(t)

(dr2

1− kr2+ r2(dθ2 + sin2 θdφ2)

), (1.1)

where a(t) is the scale factor of the universe, and the constant k is the spatialcurvature, which we will take equal to zero throughout this work.

In order to describe the dynamical evolution of the universe, we need to solveEinstein equations,

Gµν ≡ Rµ

ν −1

2δµνR = 8πGT µν , (1.2)

where Rµν , R are the Ricci tensor and Ricci scalar, respectively, T µν is the mo-

mentum energy tensor and G is the gravitational constant, related with the Planckmass MP through G = M

−1/2P , using natural unities. In Eq. (1.2) we have not

included a cosmological constant (Λ) term, due to its negligible influence during theinflationary epoch [24].

Using the metric (1.1) in Eq. (1.2), the behavior of the background can bedescribed by

H2 =8πGρ

3− k

a2,

ρ+ 3H(ρ+ p) = 0, (1.3)

where the dots represent the derivative with respect to the physical time t, H = a/ais the Hubble parameter, p is the pressure and ρ is the energy density.

The evolution of the universe depends on its contents, and also on the equationof state, relating the energy density ρ with the pressure p

p = ωρ, (1.4)

where ω = 1/3 for radiation and ω = 0 for dust.Defining

Ω ≡ ρ

ρc, with ρc ≡

3H2M2P

8π, (1.5)

the first equation in (1.3) can be written as

Ω− 1 =k

a2H2. (1.6)

Using the SC model just considered, we can describe the behavior of the universeduring the eras dominated by radiation and matter, that is to say, from 10−2 safter the big-bang until now. Some of the most important predictions of this modelare [1, 25]: the abundance of light elements, the existence of a cosmic backgroundradiation with a Planckian spectrum at some temperature different from zero, andthe explanation of gravitational lens systems.

19

Despite the success of the SC, there are some questions and problems that cannot be explained within the model, and some of them will be summarized below.

The fundamental principles of homogeneity and isotropy can not be explainedinside this framework without violating the causality principle, it is what defines theso called horizon problem.

Even more, if the universe is plane, that is, if k ≈ 0, as evidenced by observations[13], we should have Ω ≈ 1. From the dependence of the Hubble parameter on thescale factor obtained from the solution of Eqs. (1.3), it can be shown that Ω = 1 is apoint of unstable equilibrium of Eq. (1.6) for ω ≥ 0 (which is the case for radiationand matter dominated eras), due to the fact that the term on the right hand side ofEq. (1.6) always increases. This is known as the flatness problem.

Another problem with the SC concerns the abundance of topological defects, suchas magnetic monopoles, cosmic strings, or string-wall systems that may arise fromthe spontaneous breaking of symmetries assumed in the different particle physicsmodels interacting in the FRW space-time, and which are not currently observed[26].

Finally, the question about the origin of primordial fluctuations [3], the seedsfor the large structure, galaxies, clusters, and so on, that can be seen today, has nonatural explanation in this theory.

The causes of the problems previously mentioned are related to the fact thatboth radiation, p = ρ/3, and matter, p = 0, dominated universes are decelerating,with a Hubble parameter increasing faster than the wavelength of any perturbation.If a new era, characterized by an equation of state p ≈ −ρ is admitted, the universewould experience an inflationary era, as we will show in the next section.

In such a case, any problem about causality could be avoided because the universeoriginates from a causally connected region. On the other hand, it is clear that theflatness problem could be solved, because the term to the right of (1.6), would bedecreasing at first, and then will be increasing in FRW cosmology dominated byradiation or matter, until reaching its current value.

Finally, any density of unwanted relics would be diluted by inflation and theanisotropies of the CMB, the origin of the large structure would be explained byfluctuations of the field driving inflation.

1.2 Inflationary modelIn simple terms, we can describe an inflationary era as any period in which the scalefactor of the Universe, a(t), is growing at an accelerated rate a(t) > 0, which can bewritten in an equivalent way as [2]

d

dt

H−1

a< 0. (1.7)

From Friedmann equations (1.3), it can be seen that ρ+3p < 0 has to be satisfiedduring the inflationary period. As ρ > 0, then we need a negative pressure in orderto accomplish the inflationary conditions [27].

20

In order to describe an inflationary epoch in the early universe, a plethora ofinflationary models has been proposed. There are models with scalar and vectorfields to drive inflation [28, 29], scalar-multi-tensorial theories [30], or inflationarymodels appearing as the consequence of modifications to the Einstein Hilbert la-grangian (f(R) theories), wich lead to higher derivative terms in the equation ofmotion [5, 31, 32], among other possibilities. An important point to be remarkedhere is that several of these models are dynamically equivalent to theories with scalarfields driving inflation.

In particular, higher derivative theories [9, 33], the object of study of the firstpart of this work, can be mapped in an inflationary model driven by a scalar field,and for that reason we present the main features of such inflationary theory in thefollowing.

In the simplest case, the inflationary theory uses a scalar field φ minimally cou-pled to gravity1, which evolves according to an arbitrary potential function V (φ),in such a way that the lagrangian density is given by [34]:

L =1

2∂µφ∂

µφ− V (φ), (1.8)

and we can use a potential as simple as V (φ) ∝ φ2, on the condition that it has asufficiently flat region to develop the slow roll regime [35], to be defined later.

In a toy model with effective potential V (φ) = m2

2φ2 and mass m, several regimes

are possible depending on the value of the field φ: for sufficiently small values ofV (φ) small field fluctuations are generated, and this field moves slowly towardsthe minimum of its potential; while for values close to the minimum of V (φ), thescalar field undergoes rapid oscillations, creating pairs of particles and warmingup the universe. Since this potential function has a minimum at φ = 0, it could beexpected that the scalar field φ oscillates around this minimum. If the Universe doesnot expand, the equation of motion for φ in this regime is similar to the harmonicoscillator case, φ = −m2φ [36].

In the general case, a new term 3Hφ appears due to the expansion of the universe,modifying the equation of motion obtained from the Robertson-Walker metric [37]as

φ+ 3Hφ+ V ′(φ) = 0, (1.9)

which, for the special case of a quadratic potential considered before, becomes:

φ+ 3Hφ = −m2φ, (1.10)

and the term 3Hφ can be interpreted as a friction term [38].Assuming that the universe can be modelled by a perfect fluid and using Noether’s

theorem [39], we can obtain the pressure and the energy density, calculated fromthe lagrangian (1.8) as [40]

ρ = 12φ2 + V (φ), (1.11)

p = 12φ2 − V (φ), (1.12)

1That is to say, we will not include terms proportional to Rφ in the Lagrangian density, whereR is the Ricci scalar and the proportionality coefficient is a model-dependent parameter.

21

and from these expressions we can see that, if the potential energy is much largerthan the kinetic energy, we have that p ≈ −ρ, in such a way that the inflationarycondition is satisfied.

From Einstein equations it can be shown that the energy density ρ, and thereforethe Hubble parameter H, are constant in this case [38], leading to the exponentialdependence of the scale factor with time:

a(t) ≈ eHt, (1.13)

which shows that there is an accelerated expansion [35, 41] as long as φ2 V (φ)[42], which is one of the slow roll conditions, presented in the next section.

1.2.1 Slow-roll approximation

Inflation dynamics uses an approximation known as slow-roll, which assumes thatthe field is rolling slowly towards the minimum of its potential [43]

φ 3Hφ, φ2 V (φ), (1.14)

in such a way that the Einstein equation and the equation of motion for the φ fieldcan be written as

H2 ≈ 8π3M2

PV (φ), 3Hφ ≈ −V ′(φ), (1.15)

while the equation of state takes the form of p ≈ −ρ.In this case, as was shown in the previous section, the universe would perform

an accelerated expansion [35, 41]2.Two parameters enable us to describe the properties that the inflaton field po-

tential has to satisfy in order to provide a slow-roll regime3:

ε =M2

P

2

(V ′

V

)2

,

η = M2P

V ′′

V. (1.16)

Using (1.15) and (1.16) we can note that the slow roll conditions (1.14) are equivalentto [48]:

ε 1, |η| 1. (1.17)

These parameters will be used to determine observables such as the spectral indexns of the power spectrum of primordial curvature perturbations, and the tensor-to-scalar ratio r, as we will see later.

2The slow-roll approximation is not required in order to inflation to happen [44]. Potentialssatisfying 〈V − φV ′〉 > 0 can develop inflation even after a slow-roll regime [45].

3In this case we have used the slow-roll approximation on the potential (PSRA) [46], where theconditions are on the potential form. While in the slow-roll for the Hubble parameter (HSRA) [47]the conditions are on the Hubble parameter during inflation.

22

The amount of inflation is usually quantified by the number of e-foldings N(t)[44]

N(t) = ln(a(tf )

a(t)

), (1.18)

where tf is the cosmological time at the end of inflation. N(t) can be written interms of the Hubble parameter or the scalar potential as [49]

N(t) =

∫ tf

t

Hdt =

∫ φf

φ

H

φdφ ≈ − 1

M2p

∫ φf

φ

V

V ′dφ. (1.19)

In order to solve the problems of the SC, about 50 − 70 e-foldings are needed,depending on the particular model. At the end of inflation, the slow-roll conditionsare violated, ε ≈ 1 and |η| ≈ 1, and for this regime N(t) is very small [43].

1.2.2 Cosmological perturbations in inflation

During inflation, in addition to the motion of the classical field given by (1.9), thereare quantum fluctuations. Due to the fact that the energy density is dominatedby the potential energy, provided that the φ field satisfies the slow-roll conditions,fluctuations of the field generate fluctuations in the energy density [50].

For inflation driven by a scalar field, with lagrangian given by (1.8), these quan-tum fluctuations generate perturbations in both the scalar field and the metric tensor[51]

gµν −→ gµν + δgµν , φ −→ φc + δφ, (1.20)

where φc is the solution to the background equation (1.9), and gµν is the FRWmetric.

The primordial scalar (PR(k)) and tensor (PT (k)) spectra can be written in termsof the Hubble parameter as

PR(k) =

(H2k

2πφc

)2

, PT (k) =16H2

k

πM2P

, (1.21)

where the Hubble parameter Hk and the φc field are evaluated at the time when theperturbation of momentum k leaves the horizon [52].

From Eqs. (1.14) and (1.15), it is clear that the primordial spectra depend onthe scalar potential V (φc), and in such a way, they are model dependent.

Taking into account that the Hubble parameter and the φc field change slowlywith time during the inflation era, the spectra are nearly scale-invariant, that is tosay, with equal amplitudes at horizon crossing, and can be parameterized by

PR(k) = AR

(k

k∗

)ns−1

, PT (k) = AT

(k

k∗

)nT, (1.22)

where k∗ is a fiducial conformal momentum, and ns and nT are the scalar andtensor spectral indices, respectively, which can be written in terms of the slow rollparameters (1.16) [53] as

ns − 1 = 2η − 6ε, nT = −2ε, (1.23)

23

where we have used Eqs. (1.14) and (1.21).Finally, the ratio r of tensor to scalar spectra is a significant amount that allows

testing the model with the observations. For models of inflation with a scalar field,this quantity also can be written in terms of the slow-roll parameter ε as r = 16ε.

In the next chapter we are going to study one of the most appealing inflationarymodel, the R + R2 model, a particular case of the f(R) theories, first proposed byStarobinsky [5], and we will show that when the RR term is included as a smallperturbation to R+R2, the spectral index ns and the ratio tensor to scalar spectrar are consistent with experimental constraints.

24

Chapter 2

On higher derivative corrections tothe R +R2 inflationary model

As was mentioned in the previous chapter, there are a lot of inflationary models,classified mainly by the type of field driving the inflationary epoch or the scalarpotential on which this field evolves in time.

In this chapter, we will present the results published in [12]. In that paper, weanalyzed the characteristics of the R + R2 model of inflation when the term RRis included as a small perturbation.

In the previous chapter, we remarked that this kind of model leads to higherderivative terms in the equations of motion. The role of higher derivative terms canbe seen from different perspectives. In the semiclassical approach to gravity, theseterms are required to lead to a renormalizable theory [54] (see [55] for a review andfurther references).

The same situation holds in quantum gravity, where fourth derivative termsprovide renormalizability [56]. An important advantage of higher derivative termsis that their effects are strongly suppressed below the Planck scale, and hence theclassical solutions of general relativity can be seen as a very good approximationat the cosmological, astrophysical and laboratory scales. For this reason, theorieswith high derivative terms do not contradict the full bunch of experimental andobservational tests of Einstein’s gravity.

At low energies, the terms with higher derivatives can be regarded as smallperturbations of the fiducial theory of general relativity. This approach has beensuggested as an ad hoc universal solution of the ghost problem [57]. This proposalleads to the following dilemma: trying to consider all higher derivative terms asobjects to be avoided at the fundamental level and treated as perturbations, one hasto ‘forbid’ the R + R2 (Starobinsky) model of inflation [58]. It is easy to note thatthis is something difficult to accomplish. First of all, the R2-term does not produceghosts and hence there is no reason to avoid it. At the same time, this inflationarymodel is the most successful from the observational and phenomenological points ofview, so it is not easy to give it up without a real motivation. Finally, the inflationscenario requires the value of the numerical coefficient of the R2-term to be quitebig, about 5×108 [10] (see also recent work [59]). This makes the Planck suppression

25

of this term much less efficient at the scale of inflation. As a result, it would bequite natural to replace the R2-term into another side of the perturbation schemeand include it into the basic action, together with the Einstein-Hilbert term.

In this chapter we apply this idea to inflation. Namely, we add a small sixth-derivative term to the standard R + R2 action and find an upper bound on thecoefficient of the new term.

Previous attempts to analyze higher derivative corrections to the Einstein-Hilbertaction have been performed elsewhere [4, 33, 60], making a transformation of thegravitational terms to scalar fields, or with different simplifying assumptions [61, 62],which transform the higher derivative theory into a R + R2 theory with modifiedparameters. Our approach for the analysis of this higher derivative corrections, willbe to include them in the lagrangian as small perturbations, and determine theirinfluence in the parameters characterizing the cosmological perturbations, namely,the spectral index ns and the scalar-to-tensor ratio r.

This chapter is structured as follows: the R+R2 model is introduced in section2.1. In section 2.2, the RR term is included in the standard way, as a new degreeof freedom, and, finally, we present the results of treating the term RR as a smallperturbation to the Starobinsky model in section 2.3. Conclusions for this chapterare found on section 2.4

2.1 The R +R2 ModelAmong different models of inflation [3, 63, 64], the R + R2 model introduced in [5]is one of the most appealing from both theoretical and observational perspectives.It has the least number (one) of free parameters fixed by observations only. Theaction of this model is closely related to vacuum quantum corrections [65, 66] (seealso [67, 68] and [69] for recent advances in this direction) and, on the other hand, itspredictions are consistent with recent bounds including the ones set by the Planckcollaboration [13, 70, 71].

The model is described by the Einstein-Hilbert action with an extra term pro-portional to the square of the Ricci scalar R,

S0 =M2

P

2

∫d4x√−g

(R + αR2

), (2.1)

where MP is the reduced Planck mass (it is 1/√

8π times the Planck mass), α =(6M2)−1 where M is the low-curvature (|R| M2) value of the rest mass of thescalar degree of freedom (dubbed scalaron in [5]) appearing in f(R) gravity, andwe put ~ = c = 1. The theory (2.1) can be easily mapped into a metric-scalarmodel (see, e.g., [72] and [73] where the procedure is described for the general f(R)extension)

S∗0 =M2

P

2

∫d4x√−g

[φ0R− U0(φ0)

], (2.2)

where the scalar field φ0 is related to the Ricci scalar by the relation

φ0 = 1 + 2αR, (2.3)

26

and the potential function U0(φ0) is

U0(φ0) =1

4α(1− φ0)2. (2.4)

It proves useful to make a conformal transformation, introducing a new scalar fieldχ0,

gµν = gµν exp√

23χ0

MP

. (2.5)

The action which results from this procedure has a standard kinetic term, and reads

S∗0 =

∫d4x√−g

[M2P

2R− 1

2(∇χ0)2 − V (χ0)

], (2.6)

where (∇χ0)2 = gµν(∇µχ0)(∇νχ0) and V (χ0) is a potential given by the expression

V (χ0) =M2

P

8α

(1− e−

2√6

χ0MP

)2

, (2.7)

which drives the evolution of the scalar field χ0 (scalaron) and satisfies the slow rollconditions in the large field regime.

We are interested in considering the modification of the scheme described abovewhen introducing an extra term RR, treated as a perturbation. Before that, inthe next section we shall start from a brief review of the standard treatment of themodel under discussion, which implies the use of two scalar fields.

2.2 Treating RR term as a new degree of freedomIn this section we are going to summarize some of the main results obtained whenthe higher derivative term RR is taken as an extra degree of freedom. In such acase, the standard mapping is done to a theory of two scalar fields.

The new action is

S =M2

P

2

∫d4x√−g

[R + αR2 + γRR

], (2.8)

where the parameters α and γ have dimensions of [mass]−2 and [mass]−4, respec-tively. The new term is the simplest one leading to a ghost, as described in [74], andtherefore will be interesting to see how it can be treated as a small perturbation,while the ghost problem is avoided.

The action (2.8) can be written in terms of two scalar fields [9, 33]. To this endwe write the action as [33] (see also [72])

S =M2

P

2

∫d4x√−g [F (φ1, φ2) + F1(R− φ1)

+ F2(R− φ2)] , (2.9)

27

where F (φ1, φ2) = φ1 + αφ21 + γφ1φ2, F1 = ∂F

∂φ1, and F2 = ∂F

∂φ2, in such a way that

the action takes the form

S =M2

P

2

∫d4x√−g

[φ1 + αφ2

1 + γφ1φ2

+ (1 + 2αφ1 + γφ2)(R− φ1) + γφ1(R− φ2)]

=M2

P

2

∫d4x√−g

[(1 + 2αφ1 + γφ2)R− αφ2

1

+ γφ1R− γφ1φ2] . (2.10)

In order to eliminate the term with R, we integrate by parts [8],∫d4x√−gφ1R = −

∫d4x√−g∇µφ1∇µR

=

∫d4x√−gRφ1, (2.11)

and arrive to the following expression for the action

S =M2

P

2

∫d4x√−g [(1 + 2αφ1 + γφ2 + γφ1)R

− αφ21 − γφ1φ2

]. (2.12)

Defining σ = 1 + 2αφ1 + γφ2 + γφ1, the action can be written in terms of twoscalar fields in the form

S =M2

P

2

∫d4x√−g [σR + γφ1φ1 − U(φ1, σ)] , (2.13)

where U(φ1, σ) = φ1(σ − 1)− αφ21.

From Eq. (2.13) one can see that setting γ = 0, we recover the R+ αR2 case [5]in the Jordan frame.

In order to analyze the theory with two scalar degrees of freedom, we have toperform the conformal transformation of the metric, gµν = e2ϕgµν [75, 76],

S =M2

P

2

∫d4x√−ge−4ϕ

σe2ϕ

[R− 6(∇ϕ)2 − 6ϕ)

]

+ γφ1e2ϕ(φ1 − 2∇µϕ∇µφ1

)− U(φ1, σ)

. (2.14)

In this case, taking σ = e2ϕ, it is straightforward to obtain the action in the Einsteinframe

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2 − γe−2ϕ(∇φ1)2

− U(φ1, ϕ)], (2.15)

where the potential is defined as

U(φ1, ϕ) = e−4ϕ[φ1(e2ϕ − 1)− αφ2

1

]. (2.16)

28

We realize also that the action (2.15), which follows from the standard approach,has a non-standard kinetic term, of the type that were analyzed in several works[77, 78, 79, 80], always in the slow roll approximation.

One can show that the fields φ1 and ϕ satisfy the equations of motion

φ1 − 2∇µϕ∇µφ1 −e−2ϕ

2γ

[e2ϕ − 1− 2αφ1

]= 0,

φ1 +α

γφ1 = − 1

2γ(1− e2ϕ), (2.17)

which are consistent with the relation between the original fields φ1 = φ2 andreproduces the results of [9].

From Eq. (2.9), it can be seen that in this case the fields ϕ and φ1 are relatedwith the Ricci scalar R and R by

ϕ =1

2ln (1 + 2αR + 2γR),

φ1 = R, (2.18)

which are in complete agreement with the results presented in [9].From equations (2.15) and (2.16), it is clear that when γ = 0 we recover the

R + αR2 case [5] in the Einstein frame.In subsection 2.2.1 we present the derivations of the Einstein equation in the

weak field approximation up to the second order in the fields, that reproduce theresults of [9]. Also, in subsection 2.2.2, the ansatz R = r1R+r2 (proposed in [61]),where r1,2 are constants, is used to verify that the results are consistent with theones presented in [62].

2.2.1 Weak field approximation

The Einstein tensor can be found from equation (2.15), taking the variation withrespect to gµν ,

Gµν = γe−2ϕ(∇µφ1∇νφ1 −

1

2gµν∇λφ1∇λφ1

)

+ 6(∇µϕ∇νϕ−

1

2gµν∇λϕ∇λϕ

)

+1

2gµνe

−4ϕ[αφ2

1 + (1− e2ϕ)φ1

]. (2.19)

The last expression is consistent with the Einstein tensor obtained for the first timein [9], where the weak field approximation was worked out, showing that the actionin this case is given by

Swf ≈M2

P

2

∫d4x√−g

[R− 6∇λϕ∇λϕ

− γ∇λφ1∇λφ1 − 2φ1ϕ+ αφ21

]. (2.20)

29

The mixed term involving φ1 and ϕ can be removed by performing a rotation of thesefields, leading to the identification of scalar fields that can be tachyonic or physical,depending on the α and γ parameters [60]. In order to have a stable Minkowskispace it is necessary that γ < 0, within this approximation. Let us note that thiscondition does not apply within our approach to the problem.

2.2.2 Simplifying Ansatz R = r1R + r2

In this section we consider the simplifying Ansatz R = r1R + r2 proposed in [61],and later on used in [62] to analyze non-local modifications of gravity with generalform factors depending on the D’Alambertian operator applied to the Riemannand Ricci tensors and the Ricci scalar.

Our main goal is to show that the application of this simplifying Ansatz to theaction (2.15), under certain conditions for the parameters α and γ, reproduces theresults obtained in [61, 62] for this particular case.

As far as we are not considering the cosmological constant term in the action,the r2 contribution vanishes, and the ansatz becomes

φ1 = r1φ1. (2.21)

The action given by (2.15), can be written as

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2 + γφ1e

−2ϕφ1

− 2γφ1e−2ϕ∇µϕ∇µφ1 − U(φ1, ϕ)

], (2.22)

or equivalently, using Eq. (2.21), as

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2 + γr1φ

21e−4ϕ

− U(φ1, ϕ)] . (2.23)

Finally, using the relation between the fields σ = e2ϕ and φ1, we arrive at theone-scalar representation

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2 − U(ϕ)

], (2.24)

with the potential

V (ϕ) =U(ϕ)

2=

1

8(α + γr1)(1− e−2ϕ)2. (2.25)

From (2.7) and (2.25), it is clear that assuming that the scalar field ϕ evolveswithin the slow roll approximation, the scalar spectral index ns and the tensor-to-scalar ratio r, which depend on the potential, are exactly the same as obtained inR2 inflation, as was previously shown by [61] for the model R+Rn, and after them,in the case of a non-local framework. There is no change in the tensor-to-scalar

30

ratio r, because the Weyl term C2 is excluded and the non-local operator FC() isabsent [62] in our analysis.

It is important to remark that, as was shown in Refs. [61, 62], any solution of theR +R2 theory satisfying the ansatz (2.21), is also a solution of the R +R2 +RRtheory, but this is not the whole solution to the theory, as there could be solutions toR+R2+RR which do not satisfy the simplifying ansatz (2.21). On the other hand,if we assume that inflation is just as in the R+R2 model, we can take the RR termas a small perturbation (as will be done in the following section), in the same spiritin which Ref. [57] analyzes higher order terms as corrections to the Einstein-Hilbertaction, leading to the determination of new solutions of the corrected theory. In ourcase, these solutions are independent of the simplifying ansatz (2.21), but give thesame results when this assumption is taken as a particular case.

2.3 Treating RR term as a small perturbation tothe R +R2 model

In the last section we introduced the standard mapping of the R+R2 +RR theoryon an inflationary model with two scalar fields, and we learned that this model canbe stable in the weak field approximation as long as γ < 0, and that it is consistentwith the simplifying ansatz R = r1R + r2.

Nevertheless, this kind of analysis is not completely free of problems, becausethe equations of motion following from the action (2.15) do not satisfy the slowroll conditions generically, as discussed in [81]. In this situation, definition of newslow roll parameters and alternative treatments have been proposed, for instance,in [82, 83, 84].

However, the problem is that the “standard” treatment of the situation in cos-mology which was done in the references mentioned above is opposite to the onewhich is usually considered “standard” in dealing with higher derivative theories[57]. So, our goal will be to close this gap and consider the last term in Eq. (2.8)as a perturbation. The main point is that in this case we cannot use the standardscheme of mapping to the metric-scalar models (see, e.g., [72]). Instead, we have tofollow the treatment of the new term as a perturbation, meaning that the numberof degrees of freedom is not increased, in contrast to the action in Eq. (2.15).

Treating the RR-term as a perturbation, one can suppose that in mappingto a scalar-metric model, the term RR should be substituted by R(φ0)R(φ0),where φ0 is a scalar field, similar to φ0 in Eq. (2.2). The action is perturbed by theinclusion of the γ term,

S = S∗0 + Sγ, (2.26)

where Sγ is defined as

Sγ =M2

Pγ

2

∫d4x√−gRR

R=

(φ0−1)2α

=M2

Pγ

8α2

∫d4x√−gφ0φ0. (2.27)

31

Let us note that the relation between R and φ0 in this formula is exactly the sameas the obtained for the unperturbed R+R2 model given by equation (2.3) withoutany changes. This procedure means that we disregard all possible terms of higherorders in γ.

In order to obtain the scalar mapping under this approximation, we use therelation between the σ = e2ϕ and φ1 fields given by the equation of motion (2.17),taken up to linear order in the γ

2α operator. This leads to

φ1 =1

2α(e2ϕ − 1)− γ

2α2e2ϕ. (2.28)

When Eq. (2.28) is substituted back into the action, this gives us

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2

(1 + ke2ϕ

)− U(ϕ)

],

where k =γ

6α2and U(ϕ) =

e−4ϕ

4α(e2ϕ − 1)2 (2.29)

which has exactly the same form as the one in Eq. (2.25). Nevertheless, in this casethe kinetic term has a non-canonical form. Thus, to find the mass of the field in theMinkowski limit, it has to be transformed into the standard form.

It is important to emphasize that all the analyses made in this work are concernedwith the case |k| 1, in order to allow the treatment of the R term as a smallperturbation to the R +R2 theory.

It is easy to see that when the Ansatz (2.21) is used alongside with the previousapproximation, we recover the results of [61, 62]. In order to check this, we writethe action (2.29) in the equivalent way,

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2 + 3k(σ − 1)ϕ

− U(ϕ)], where σ = e2ϕ, (2.30)

as defined above. Now, we can use Eq. (2.28) to find the relation(

1− γ

α)−1 [

r1 +σ − γ

α(σ)

]= r1σ. (2.31)

When expanded up to the linear order in the operator γα we arrive at the relation

σ = r1(σ − 1).With these considerations, the action (2.30) becomes

S =M2

P

2

∫d4x√−g

[R− 6(∇ϕ)2 − U(ϕ)

], (2.32)

where the potential is defined as

U(ϕ) =α− r1γ

4α2e−4ϕ(1− e2ϕ)2. (2.33)

32

Comparing the last expression with the potential corresponding to the nonperturbedStarobinsky model in the Einstein frame given by Eq. (2.7), one can see that thevalues of the spectral index ns and the tensor-to-scalar ratio r will not be modified,which is consistent with the results for these quantities in [62] based on the use ofthe specific Ansatz (2.21) .

Let us stress that although the term γRR studied here is a particular caseof the F () analyzed in previous works, in our case the simplifying Ansatz (2.21)is not an additional requirement, but only a special case. Thus, our study in thiswork is not a particular case of that in [61, 62], but represents a qualitatively newapproach to the RR term in the action.

The bilinear part of the action (2.30) can be cast into the form

S =

∫d4x√−g

M2P

2R− 1

2(∇χ)2 − 1

2M2

χχ2, (2.34)

where fields χ and ϕ are related by the relation

ϕ =χ√

6 (1 + γ/6α2)MP

. (2.35)

Let k ≡ γ6α2 . If |k| 1, the mass of the field χ is

M2χ ≈M2 − γ

36α3, (2.36)

where M2 (defined below Eq. (2.1)) is the scalaron mass in the Starobinsky model.From Eq. (2.36) one can see that the mass of the scalar χ can be greater or smallerthan M depending on the sign of the parameter γ. This behavior is preserved bysmall values of |k| 1 in the general case of (2.29), as it will be seen from theestimated values for this quantity obtained using the Planck collaboration results[13].

Returning to the general expression for the action (2.29), the field transformationthat turns the kinetic term in a canonical form should satisfy

(dχ

dϕ

)2

= 6M2P

(1 + ke2ϕ

), (2.37)

and then the action becomes

S =

∫d4x√−g

M2P

2R− 1

2(∇χ)2 − V (ϕ(χ))

. (2.38)

In the Einstein frame, the potential is given by

V (χ) = V (ϕ(χ)) =M2

P

8α

(1− e−2ϕ(χ)

)2, (2.39)

where the dependence of the intermediate field ϕ on the scalar field χ in Eq. (2.39)(which may be called new scalaron) can be obtained by solving the transcendentalequation that follows from (2.37),

χ√6MP

= ln

[1−√

1 + ke2ϕ

eϕ(1−√

1 + k)

]+√

1 + ke2ϕ −√

1 + k. (2.40)

33

0

0.5

1

1.5

2

2.5

3

3.5

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2

2.5

3

3.5

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

χ√6M

P

ke2

ϕ≈1

k

0

0.5

1

1.5

2

2.5

3

3.5

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

χ√6M

P

ke2

ϕ≈1

k

0

0.5

1

1.5

2

2.5

3

3.5

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Figure 2.1: The energy scale of the new scalaron χ as a function of the perturbativeparameter k. The shaded regions are excluded by the constraints imposed by Planckdata [13], as will be shown later.

In the last expression for the particular case ke2ϕ ≈ 1, and taking into accountthat |k| 1, but k 6= 0 we have

χ√6MP

∣∣∣∣ke2ϕ≈1

≈ −1

2ln |k| − k

4+

1

4Sign(k), (2.41)

which shows that a large field inflation can be developed when |k| 1, as can beseen from figure 2.1, where we have plotted Eq. (2.41) which shows the behaviorof the new scalaron χ in the particular case ke2ϕ ≈ 1, where the shaded regionsare excluded by the Planck data, as will be discussed below. Let us note that thecondition ke2ϕ ≈ 1 has been chosen since it corresponds to the maximal value ofthe parameter k which is compatible with our approximation |k| 1. For smallervalues of |k| the effect of the RR term will be less significant.

Let us note that the general expression (2.40) can be used as the basis of a metric-scalar cosmological model even for the values of k which do not satisfy the condition|k| 1. However, in this case there is no direct link with the main perturbativeapproach for the R +R2 +RR theory, which we aim to develop in this chapter.

In Fig. 2.2, the plot of the potential V (χ) is shown for different values of theparameter k, with k = 0 corresponding to the R+R2 model, and the other values tothe extra RR term with different coefficients. One can observe that the presence ofthe RR term changes the shape of the potential including its flatness. For k > 0,the slow-roll inflationary regime ends for slightly larger values of the scalar field,implying that for larger values of k, inflation happens at higher energy scales thanfor the standard R +R2 model.

For positive k, all expanding spatially-flat FRW universes evolve to the dust-likeone filled by massive scalarons at rest at late times, like in the case of the Starobinskymodel. As we will see later, at earlier times, they can develop inflation in the slowroll regime. For the case of negative values of k (remember |k| 1), there is a

34

maximum in the potential for a critical value of the χ field, given by the expression

χmax√6MP

= ln

( √|k|

1−√

1− |k|

)−√

1− |k|, (2.42)

which comes from the constraint of a real χ field in Eq. (2.40). In this case, slowroll inflation can take place only for values of k which are close to zero, as will beseen from the behavior of the slow roll parameters ε and η. In fact, this is not toorelevant, since all our analysis is valid only for small values of the parameter |k|.

The nonzero k or, equivalently, the γ term, modifies the value of χ in which thelast 60 e-folds of inflation begin, leading to a modification of observable parameterssuch as the tilt of the primordial power spectrum of scalar (adiabatic) metric per-turbations ns and the scalar-to-tensor ratio r, as we will see below. Furthermore,the RR-type perturbation modifies the symmetry of the scalar potential near itsminimum, and that can affect the oscillations of the field χ (new scalaron) afterinflation and gravitational creation of particles and antiparticles by these oscilla-tions through parametric resonance [5, 85, 86, 87]. The next important question iswhether a non-zero γ modifies the conditions of a slow roll inflation.

8αM2

PV (χ)

χ√6MP

k = −0.1k = −0.07k = 0.0k = 0.25k = 0.5

Figure 2.2: RR correction to the Starobinsky model of inflation for different valuesof the parameter k. The black continuous line corresponds to the original R + αR2

model, and the colored dashed lines to nonzero values of k.

2.3.1 Slow-roll conditions

As far as the model with non-zero γ-term is mapped into a single scalar field action,the analysis of the slow roll conditions can be performed in a standard way.

In order to let inflation last for a sufficient amount of time, the time derivativeof the Hubble parameter H has to be sufficiently small. As a result, the slow rollparameters

ε = − H

H2, η = ε− ε

2εH= − χ

Hχ, (2.43)

35

have to be much smaller than one (in the case of the parameter η, its modulus),leading to a negligible contribution of the kinetic energy of the field during inflation.

Using Friedmann equations, one can express the slow roll parameters in termsof the potential as

ε =M2

P

2

[V ′(χ)

V (χ)

]2

and η = M2P

V ′′(χ)

V (χ)(2.44)

and, using Eq. (2.40), the two parameters can be written in terms of the field ϕ,

ε =4

3

1

(1 + ke2ϕ) (1− e2ϕ)2 , (2.45)

η = −4

3

[e−2ϕ (1− 2e−2ϕ) + k

2(3− 5e−2ϕ)

(1 + ke2ϕ)2 (1− e−2ϕ)2

], (2.46)

from which we can see that in the limit k = 0 the slow roll parameters of the R+αR2

model are recovered.For small values of k > 0, one can see that the slow roll conditions are satisfied

for large enough fields, while for k < 0 there is a maximum value of the χ field wherethe slow roll regime is valid: when |k| 1. For k < 0, we have a slow roll regimefor a wide range of the field, as shown in Fig. 2.3.

The number of inflationary e-folds N in the Einstein frame is given by

N(χ) =1

M2P

∫ χ

χe

V (χ)

V ′(χ)dχ, (2.47)

where χe corresponds to the end of inflation. In terms of the field ϕ we get

N(ϕ) =3

4

[− 2ϕ+ (1− k) e2ϕ +

k

2e4ϕ]∣∣∣∣ϕ

ϕe

, (2.48)

which gives the standard results for the R + αR2 model for k = 0.Assuming that χe χ (that is equivalent to ϕe ϕ in the case of large field

inflation), one can neglect the linear terms in ϕ in Eq. (2.48), which boils down to

N(ϕ) ≈ 3

4

[(1− k)e2ϕ +

k

2e4ϕ

]. (2.49)

One can use this relation to derive the value ϕN of the field ϕ, corresponding tothe instant when the universe expanded by N e-folds,

ϕN ≈1

2ln

[1− 1

k±√(

1− 1

k

)2

+8N

3k

], (2.50)

where the positive sign has to be taken in order to recover the results for the R+R2

model. Taking into account that |k| 1 we can simplify this expression as

ϕN ≈1

2ln

[−1

k+

√1

k2+

8N

3k

]. (2.51)

36

0

0.2

0.4

0.6

0.8

1

0.4 0.5 0.6 0.7 0.8 0.9 1

Slow

rollparam

eter

ǫ

χ√6MPl

k = −0.1k = −0.07k = 0.0k = 0.25k = 0.5

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Slow

rollparam

eter

η

χ√6MP

k = −0.1k = −0.07k = 0.0k = 0.25k = 0.5

Figure 2.3: The slow roll parameters ε (up) and η (down) for the model with RRterm as functions of the inflaton field χ.

37

When the slow roll conditions are satisfied, the amplitude of scalar (curvature)and tensor perturbations can be written in terms of the potential V (χ) and itsderivatives at the moment when their physical wavelength λ = p/a(t), p = constcrosses the Hubble radius H−1(t) during inflation. The same matching conditionhelps to express N as a function of the present physical scale λ = a(t0)/p wheret0 is the present moment. Then one gets the standard expressions for the spectralindex ns(p) of the power spectrum of primordial curvature perturbations and thetensor-to-scalar ratio r(p) in the leading order of the slow-roll approximation:

ns − 1 = −6ε+ 2η = M2P

[2V ′′

V− 3

(V ′

V

)2], (2.52)

r = 16ε = −8M2P

(V ′

V

)2

. (2.53)

Because of the conformal transformation between the Jordan and Einstein frames,the same value of a perturbation as a function of N and, finally, p corresponds tosomewhat different physical scales in the Jordan and Einstein frames. Since thestandards of length and time intervals are defined in Jordan frame which can beconsidered as the physical one from the measurement point of view, the number ofe-folds in the Jordan frame NJ is more directly related to observations. However,the difference between N and NJ is small, of the order of the next correction to theslow-roll approximation (less than a few percent for the model in question), so wemay neglect it in the leading order.

Using equations (2.45) and (2.46), we can obtain the following analytical expres-sions for ns and r as functions of k and the number of e-folds N:

ns − 1 =9k [18k3 + 6k2(Sqk,N + 12N − 29)]

(Sqk,N − 3)2(Sqk,N + 3k)2+

9k2 [16N(Sqk,N + 4N − 18)− 52(Sqk,N + 261)]

(Sqk,N − 3)2(Sqk,N + 3k)2+

9k [8N(9− 2Sqk,N) + 40Sqk,N − 138]− 54Sqk,N + 18

(Sqk,N − 3)2(Sqk,N + 3k)2,

r =576k2

(Sqk,N − 3)2(Sqk,N + 3k)2, (2.54)

where we have defined Sqk,N =√

24kN + 9(k − 1)2.From equations (2.54), we can realize that when k = 0, we recover the predictionsof the R +R2 model [10, 88],

ns − 1 ≈ − 2

Nr ≈ 12

N2, (2.55)

and up to the order kN there is no shift in ns − 1 and r, and their corrections areof order k/N or k/N2, respectively.

38

+

+ +

>

<

=

=

0.94 0.96 0.980.

0.05

0.1

Primordial tilt (ns)

Tensor-to-scalarratio(r)

Figure 2.4: Observational constraints set by the Planck collaboration [13, 71] on thescalar-to-tensor ratio r and scalar spectral index ns, and the prediction for thesequantities in the R + R2 + RR model. The allowed regions for this model arepresented in red and green color for negative and positive values of the k parameter,respectively.This figure is an original result of this thesis and appears in [12].

The comparison of this inflationary model with the observational constraints setby the Planck collaboration [13, 71] is illustrated in Fig. 2.4. The last Planck dataconstrain these quantities as

ns = 0.9649± 0.0042, r < 0.10. (2.56)

In Fig. 2.4 we show the Planck constraints on the values of ns and r, and theprediction for this quantities in the R+αR2 + γRR model. This figure shows the68% (dark blue and dark yellow) and 95% (light blue and light yellow) C.L. regionsfor the measurements of r and ns, taking the combined data as stated in the plotkey, and the variation of the R + αR2 model (black line) due to the inclusion ofthe γ term regarded as a small correction for k > 0 and k < 0 in green and red,respectively.

Using a numerical routine, from Fig. 2.4 we have found the maximum positivevalue of k in order to keep the predictions of the RR model inside the 68% C.L.region, concluding that, for N = 50 and N = 55 e-folds, any value of k satisfyingthe perturbative condition |k| 1 keeps the predictions of the RR model insidethe 68% C.L. region. For the case N = 60 the maximum value of k that satisfiesthis condition is kmax ≈ 0.30. On the other hand, for negative values of k, Planck

39

+

+ α + γ

-0.02 0.0.

0.008

0.016

ηV

ϵV

Concave Convex

Figure 2.5: Observational constraints on the slow roll parameters set by Planckcollaboration [89], and the prediction for these quantities in the R+ αR2 + γRR.The subindex V indicates that the slow roll parameters are written as a function ofthe scalar potential.This figure is an original result of this thesis and appears in [12].

results constraints the minimum value of k to −0.0060, −0.0052 and −0.0045, whenN = 50, 55 and 60, respectively.

Furthermore, Fig. 2.5 shows the Planck constraints [71, 89] for the relation be-tween the slow roll parameters (2.44), at the 68% (dark blue) and 95% C.L. (lightblue). Also in this figure one can see the prediction of our present model (light greenfor k > 0 and red for k < 0).

Finally, one can check how the upper bound for the energy scale M may changein the case of the R + R2 + RR model. Using a procedure similar to the onepresented in [59], we use the expression (2.39) for the inflationary potential andEq. (2.45) for the first slow roll parameter, written in terms of the number of e-foldsN , from which we get the curvature perturbations

∆2R =

1

M2P

√8V

3ε, (2.57)

and make a comparison with the last Planck results [13]. For the case of 60 e-foldsand using the value of k which assures that the correction due to the RR termlies inside the 68% C.L. regions of both figures 2.4 and 2.5 (hence k ≈ 0.30), themaximum value of M would be

Mmax = (2.21± 0.01)× 10−5MP , (2.58)

which is consistent with the result of [10].

2.4 ConclusionsWe considered an example of treating the extension of R+R2 inflationary model byadding a small perturbation of the form RR. Treating the term with higher thanfour derivatives as a small perturbation means that one can perform the mappingof this modified gravity theory to the scalar-tensor model with one scalar only, and

40

this opens the way for easy and explicit analysis of the observational constraints onthe extra terms.

The case which considered here provides an especially simple mapping procedure,but there is a good chance that the same result can be achieved for any other formof extension in the action.

Part II

Axions in the 3− 3− 1 model

41

42

Chapter 3

Axion dark matter in the 3− 3− 1model

Observations in a wide range of scales point to the existence of a kind of matterwith no direct interaction with light, making about 85% of the total mass of theuniverse. This dark matter (DM) is one of the greatest puzzles in modern science,alongside with dark energy, the leading component of the universe [13].

Although there are currently many candidates for dark matter, one of the mostcompelling candidates to DM is the weakly interacting massive particle (WIMP),a stable (or very long lived) particle with an interaction cross section in the elec-troweak range [90]. These particles have evaded the latest efforts of direct detectionexperiments [91], claiming for different candidates to DM.

Axions, which initially arise as a consequence of the spontaneous breaking ofa U(1) symmetry assumed in the Peccei-Quinn (PQ) mechanism used to solve thestrong CP problem, could be a possible candidate for DM [92], as long as the PQsymmetry breaking scale is sufficiently larger than the electroweak scale.

Due to the presence of instantons, the QCD lagrangian has a new term that isnot naturally P and T invariant, and because any theory has to be CPT invariantthe QCD is not CP-invariant. This is known as the strong CP problem. To solvethis, the PQ mechanism was proposed by R. D. Peccei and Helen R. Quin in 1977[93], they showed that when the lagrangian of the QCD has a U(1) symmetry whichis broken only by the axial anomaly, the CP-conservation is automatically satisfied.

The current experimental observations strongly constrain the abundance of DM.In the case of axions, its relic abundance is due to the misalignment mechanism andthe decay of axionic strings and domain walls, topological defects produced due tothe spontaneous breaking of the PQ symmetry.

If the PQ-symmetry breaking was prior to inflation, the effects of topologicaldefects are washed out [94], as discussed in the first part of this work. On thecontrary, if the PQ-breaking happened after inflation these defects can increase theabundance of axions, as we will show later.

Whatever the scenario, it is clear that physics beyond the standard model (SM)of particle physics is necessary. Among a plethora of frameworks beyond SM, a wayof introducing new physics is to consider a model with a larger symmetry group

43

[95, 96]. In particular, there is a class of models based on the SU(3)C ⊗ SU(3)L ⊗U(1)X gauge group (the so called 3 − 3 − 1 models, for shortness) [97], which areinteresting extensions of the SM. This model has many appealing features, but themost important for this work is that it admits the introduction of a PQ symmetry,resulting in an invisible axion in the spectrum [98], that could explain the darkmatter puzzle.

Taking this into account, this chapter will be devoted to the analysis of axionDM in the framework of the 3− 3− 1 model, analyzing the possible mechanisms forits production and its prospects of detection.

The content of this chapter is as follows. In Sec. 3.1 we summarize the generalfeatures of the 3 − 3 − 1 model, including its matter content, Yukawa interactionsand scalar potential. After that, in Sec. 3.2, we show the main steps to make theaxion invisible and the PQ mechanism stable against gravitational effects, and wealso show the axion effective potential from which its mass is derived, we follow [99],and refer the lector to this reference and the references there for more details. Then,in Sec. 3.3, we consider the axion production mechanisms in order to compute itsabundance in the Universe, this section is based in [100]. Results for the vacuummisalignment and decay of the string and string-wall system mechanisms for theproduction of axions are given. Following that, in 3.4, we confront the predictionsfrom the previous section with observational constraints, coming mainly from thePlanck collaboration [13] results for the DM abundance, the neutron electric dipolemoment (NEDM) data [101] and direct axion searches [102, 103, 104, 105], in orderto constrain the parameter space of the model. Finally, in Sec. 3.5, we comparethree 3−3−1 models including a PQ symmetry, in which different number of domainwalls and discrete symmetries appear, and present our results concerning the darkmatter abundance, finding constraints on the parameter space of the models, takinginto account observations on the NEDM, gravitational effects on the mass of theaxion and prospects of observation of axion DM. Conclusions for this chapter arepresented in 3.6.

3.1 A brief review of the modelWe consider the 3 − 3 − 1 model with right-handed neutrinos, Na, in the samemultiplet as the SM leptons, νa and ea. In other words, in this model all of theleft-handed leptons, FaL = (νa, ea, N

ca)

TL with a = 1, 2, 3, belong to the same

(1, 3, −1/3) representation, where the numbers inside the parenthesis denote thequantum numbers of SU(3)C , SU(3)L and U(1)X gauge groups, respectively. Thismodel was proposed in Refs. [106, 107] and it has been subsequently considered inRefs. [95, 108, 109, 110, 111, 112, 113, 114, 115, 116]. It shares appealing featureswith other versions of 3−3−1 models [96, 97, 117, 118, 119, 120, 121, 122]. Further-more, the existence of right-handed neutrinos allows mass terms at tree level, but itis necessary to go to the one-loop level to obtain neutrino masses in agreement withexperiments [111].

The remaining left-handed fermionic fields of the model belong to the following

44

representations

Quarks: QL = (u1, d1, u4)TL ∼ (3, 3, 1/3) , (3.1)

QbL = (db, ub, db+2)TL ∼ (3, 3, 0) , (3.2)

where b = 2, 3; and “∼” means the transformation properties under the local sym-metry group. Additionally, in the right-handed field sector we have

Leptons: eaR ∼ (1, 1, −1) , (3.3)Quarks: usR ∼ (3, 1, 2/3) , dtR ∼ (3, 1, −1/3) , (3.4)

where a = 1, 2, 3; s = 1, . . . , 4 and t = 1, . . . , 5.In order to generate the fermion and boson masses, the SU(3)C⊗SU(3)L⊗U(1)X

symmetry must be spontaneously broken to the electromagnetic group, i.e., to theU (1)Q symmetry, where Q is the electric charge. To do this, it is necessary tointroduce, at least, three SU(3)L triplets, η, ρ, χ, as shown in Ref. [115], which aregiven by

η =(η0

1, η−2 , η

03

)T ∼ (1, 3, −1/3) , ρ =(ρ+

1 , ρ02, ρ

+3

)T ∼ (1, 3, 2/3) , (3.5)

χ =(χ0

1, χ−2 , χ

03

)T ∼ (1, 3, −1/3) . (3.6)

Once these fermionic and bosonic fields are introduced in the model, we canwrite the most general Yukawa Lagrangian, invariant under the local gauge group,as follows

LYuk = LρYuk + LηYuk + LχYuk, (3.7)

with

LρYuk = αtQLdtRρ+ αbsQbLusRρ∗ + Yaa′εijk

(FaL)i(Fa′L)cj (ρ∗)k + Y′aa′FaLea′Rρ

+H.c., (3.8)LηYuk = βsQLusRη + βbtQbLdtRη

∗ + H.c., (3.9)LχYuk = γsQLusRχ+ γbtQbLdtRχ

∗ + H.c., (3.10)

where εijk is the Levi-Civita symbol and a′, i, j, k = 1, 2, 3 and a, b, s, t are in thesame range as in Eq. (3.3). It is also straightforward to write down the most generalscalar potential consistent with gauge invariance and renormalizability as

V (η, ρ, χ) = VZ2 (η, ρ, χ) + VZ2(η, ρ, χ) ; (3.11)

withVZ2 (η, ρ, χ) = −µ2

1η†η − µ2

2ρ†ρ− µ2

3χ†χ

+λ1

(η†η)2

+ λ2

(ρ†ρ)2

+ λ3

(χ†χ)2

+ λ4

(χ†χ) (η†η)

+λ5

(χ†χ) (ρ†ρ)

+ λ6

(η†η) (ρ†ρ)

+ λ7

(χ†η) (η†χ)

+λ8

(χ†ρ) (ρ†χ)

+ λ9

(η†ρ) (ρ†η)

+ [λ10

(χ†η)2

+ H.c.]; (3.12)VZ2

(η, ρ, χ) = −µ24χ†η

+λ11

(χ†η) (η†η)

+ λ12

(χ†η) (χ†χ)

+ λ13

(χ†η) (ρ†ρ)

+λ14

(χ†ρ) (ρ†η)

+ λ15εijkηiρjχk + H.c. (3.13)

45

We have divided the total scalar potential V (η, ρ, χ) in two pieces, VZ2 (η, ρ, χ), in-variant under the Z2 discrete symmetry (χ→ −χ, u4R → −u4R, d(4,5)R → −d(4,5)R,and all the other fields even by the symmetry), and VZ2

(η, ρ, χ), which breaks Z2.This discrete symmetry is motivated by the implementation of the PQ mechanismas shown below.

It is well known that the minimal vacuum structure needed to give masses forall the particles in the model is

〈ρ〉 =1√2

(0, vρ02 , 0

)T, 〈η〉 =

1√2

(vη01 , 0, 0

)T, 〈χ〉 =

1√2

(0, 0, vχ0

3

)T, (3.14)

which correctly reduces the SU (3)C ⊗ SU (3)L ⊗ U (1)X symmetry to the U (1)Qone. In principle, the remaining neutral scalars, η0

3 and χ01, can also gain VEVs.

However, in this case, dangerous Nambu-Goldstone bosons can arise in the physicalspectrum, as shown in Ref. [123]. In this work, we are going to consider only theminimal vacuum structure given in Eq. (3.14).

3.2 Implementing a gravity stable PQ mechanismin the model

The key ingredient to implement the PQ mechanism is the invariance of the entireLagrangian under a global U (1) symmetry, called U (1)PQ, which must be bothafflicted by a color anomaly and spontaneously broken [124, 125, 126]. In general,the implementation of the PQ mechanism in the 3 − 3 − 1 models is relativelystraightforward [95, 113, 127]. In particular, in Ref. [95] a gravitationally stable PQmechanism for the model considered here is successfully implemented. We are goingto review its main results for completeness.First of all, we search for all U (1) symmetries of the Lagrangian given in Eqs. (3.7)and (3.11). Doing so, we find only two symmetries, U(1)X and U(1)B, which clearlydo not satisfy the two minimal conditions required for the U (1)PQ symmetry. SeeTable 1 for the quantum number attributions to the fields for these symmetries. Inother words, the U(1)PQ is not naturally allowed by the gauge symmetry.

Table 3.1: The U(1) symmetry charges in the Lagrangian given by Eqs. (3.7) and(3.11).

QL QiL (uaR, u4R) (daR, d(4,5)R) FaL eaR ρ (χ, η)U(1)X 1/3 0 2/3 −1/3 −1/3 −1 2/3 −1/3U(1)B 1/3 1/3 1/3 1/3 0 0 0 0

However, if the Lagrangian is slightly modified by imposing a Z2 discrete symme-try such that χ→ −χ, u4R → −u4R, d(4,5)R → −d(4,5)R, all terms in VZ2

(η, ρ, χ) areforbidden. In addition, the Yukawa Lagrangian interactions given in Eqs. (3.8-3.10)

46

are slightly modified to

LρYuk = αaQLdaRρ+ αbaQbLuaRρ∗ + Yaa′εijk

(FaL)i(FbL)cj (ρ∗)k + Y′aa′FaLea′Rρ+

H.c., (3.15)LηYuk = βaQLuaRη + βbaQbLdaRη

∗ + H.c., (3.16)LχYuk = γ4QLu4Rχ+ γb(b+2)QbLd(b+2)Rχ

∗ + H.c. . (3.17)

Consequently, a U(1)PQ symmetry is automatically introduced, see Table 3.2.

Table 3.2: The U(1)PQ charges in the model with a Z2 discrete symmetry such thatχ→ −χ, u4R → −u4R, and d(4,5)R → −d(4,5)R .

QL QiL (uaR, u4R) (daR, d(4,5)R) FaL eaR ρ (χ, η)U(1)PQ −2 2 0 0 1 3 −2 −2

As η, ρ, χ get VEVs, an axion appears in the physical spectrum. However, it isa visible axion 1 because the U(1)PQ symmetry is actually broken by vρ02 , which isupper bounded by the value of vSM ' 246 GeV, as shown in Refs. [95, 123]. Hence,this scenario is ruled out [129]. Nevertheless, a singlet scalar, φ ∼ (1, 1, 1), can beintroduced in order to make the axion invisible. Its role is to break the PQ symmetryat an energy scale much larger than the electroweak one. This field does not coupledirectly to quarks and leptons, but acquires a PQ charge by coupling to the scalartriplets. With the PQ charges given in the Table 3.2, the scalar φ acquires a PQcharge equal to 6 by its coupling to the η, ρ, χ scalar triplets through the interactionterm λPQε

ijkηiρjχkφ [127]. Notice that this term is allowed as long as the φ field isodd under the Z2 symmetry, i.e., Z2 (φ) = −φ.

Although the Z2 discrete symmetry apparently introduces the PQ mechanism inthe model, there are two issues with it. First, the Z2 and gauge symmetries allowsome renormalizable terms in the scalar potential, such as φ2, φ4, ρ†ρφ2, η†ηφ2,χ†χφ2, that explicitly violate the PQ symmetry in an order low enough to make thePQ mechanism ineffective. Second, since the PQ symmetry is global, it is expectedto be broken by gravitational effects. Thus, a mechanism to stabilize the axionsolution has to be introduced. As usual, the entire Lagrangian is considered to beinvariant under a ZD discrete gauge symmetry (anomaly free) [95, 113, 130, 131, 132?] and, in addition, this symmetry is supposed to induce the U(1)PQ symmetry. ForZD≥10 it is found that all effective operators of the form φN/MN−4

Pl (where N ≥ Dis a positive integer and MPl is the reduced Planck mass) that can jeopardize thePQ mechanism are suppressed. In particular, in Ref. [95] two different symmetries,Z10 and Z11, were found to stabilize the PQ mechanism for the Lagrangian givenby Eqs. (3.11,3.15-3.17) (except for the term λ15εijkηiρjχk in the scalar potential).The specific charge assignments for these symmetries are shown in Table 3.3.

In both cases, the axion, a (x), is the phase of the φ field, i.e., φ (x) ∝ exp(ia (x) /fa

),

which implies fa ≈ vφ. To make the axion compatible with astrophysical and cosmo-logical considerations, the axion-decay constant fa (related to fa by fa = fa/NDW ,

1The axion was called visible in the sense that it gives some predictions from experiments asthe decay of kaons [128].

47

Table 3.3: The charge assignment for ZD that stabilizes the PQ mechanism in theconsidered 3− 3− 1 model.

QL QiL (uaR, u4R) (daR, d(4,5)R) FaL eaR ρ (χ, η) φZ10 +7 +5 +1 +1 +7 +1 +6 +6 +2Z11 +7 +6 +1 +1 +8 +2 +6 +6 +4

with NDW being the number of domain walls in the theory), must be in the range109 GeV . fa . 1014 GeV [133](we are assuming a post-inflationary PQ symmetrybreaking scenario). Note that this high value of fa ≈ vφ vρ02 , vη01 , vχ0

3, justifies the

approximation in the form of axion eigenstates. Note that v2ρ02

+ v2η01

= v2SM and vχ0

3

is expected to be at the TeV energy scale.Now, we can go further calculating the axion mass, ma. In this model, the axion

gets mass because the U(1)PQ symmetry is both anomalous under the SU(3)C groupand explicitly broken by gravity induced operators, gφN/MN−4

Pl (with g = |g| exp iδ).These operators have a high dimension (N ≥ 10) because of the protecting Z10 orZ11 discrete symmetries. These two effects induce an effective potential for theaxion, Veff, from which it is possible to determine the axion mass.In more detail, as the U(1)PQ symmetry is anomalous, we will have a VPQ term inthe effective potential, which can be written as [134]

VPQ = −m2πf

2π

[1− 4mumd

(mu +md)2 sin2

(a (x)

2fa

)]1/2

, (3.18)

where mπ ' 135 MeV and fπ ' 92 MeV are the mass and decay constant of theneutral pion, respectively; mu and md are the masses of the up and down quarks.Note that VPQ has a minimum when 〈a (x)〉 /fa = 0, which solves the strong CPproblem in the usual way. However, because the PQ symmetry is also explicitlybroken by gravity effects, the effective potential gets another term, Vgravity, whichreads [135, 136, 137]

Vgravity ' −|g| vNφ

2N/2−1MN−4Pl

cos

(N a (x)

fa+ δD

), (3.19)

where N = 10, 11 for Z10 and Z11, respectively. The phase δD inside the trigono-metric function can be written as

δD = δ −Nθ, (3.20)

where δ is the phase of the g coupling constant (g = |g| exp iδ) and θ is the parameterwhich couples to the gluonic field strength and its dual. This extra term in the scalarpotential, Eq. (3.19), has two important consequences. First, it induces a shift inthe value of 〈a(x)〉

fawhere Veff has a minimum. Expanding Veff = VPQ + Vgravity in

powers of 〈a(x)〉fa

, we find that in the minimum, the axion VEV satisfies

| 〈a (x)〉 |fa

∣∣∣∣min

'N |g|NN−1

DW

2N2 −1

(faMPl

)N−2

M3Pl sin δD

m2πf

2π

f2a

mumd(mu+md)2

+N2|g|NN−2

DW

2N2 −1

(faMPl

)N−2

M2Pl cos δD

, (3.21)

48

where we have used vφ ≈ fa = NDWfa. Note that for |g| = 0 (or for δD = 0)we have that 〈a(x)〉

fa= 0 in the minimum, as it should be to solve the strong CP

problem. However, in the general case, the value of 〈a(x)〉fa

does not satisfy the NEDMconstraint [101], which imposes [138]

〈a (x)〉fa

= θ . 0.7× 10−11. (3.22)

In addition, Vgravity brings a mass contribution for the axion, ma, gravity. FromEq. (3.19) we obtain

m2a, gravity =

N2 |g|NN−2DW

2N2−1

(faMPl

)N−2

M2Pl cos δD. (3.23)

This contribution can, in general, be much larger than the well-known axion-massterm coming from the QCD non-perturbative terms,

m2a, QCD =

m2πf

2π

f 2a

mumd

(mu +md)2 . (3.24)

Thus, in order to maintain the axion mass stable, we are going to look for values ofthe parameters |g|, fa and δD for N = 10, 11 that both satisfy the NEDM constraintand leave the axion mass stable (ma,QCD & ma, gravity).

3.3 The Production of Axion Dark MatterCold dark matter in the form of axions can be produced by three different pro-cesses: the misalignment mechanism [139], where the axion field oscillates about theminimum of its potential, trying to decrease the energy after the breaking of thePQ symmetry; and the decay of one-dimensional and two-dimensional topologicaldefects, which appear after breaking this symmetry: global strings [140] and domainwall systems [141], respectively.

In this section we present a brief summary of the axion production mechanisms,following [134], and refer the lector to this reference and references therein for moredetails.

3.3.1 Misalignment mechanism

The equation of motion for the axion field a in a homogeneous and isotropic Universe,described by the Friedmann-Robertson-Walker (FRW) metric, is written as

a+ 3Ha+m2aa = 0, (3.25)

where H is the Hubble parameter and ma is the axion mass, and both are dependenton the temperature T.Taking into account the nonperturbative effects of QCD at finite temperature, and

49

considering the instanton system as a dilute gas, Monte Carlo simulations have beenperformed in order to fit a power law for the axion mass as a function of temperature,leading to the relation [134, 142]

m2a(T ) = cT

Λ4QCD

f 2a

(T

ΛQCD

)−n, (3.26)

where the values of the parameters are cT = 1.68 × 10−7, n = 6.68 and ΛQCD =400 MeV. Eq. (3.26) is valid in the regime where the axion mass at temperature Tis less than its value at temperature zero, given by

m2a(0) = c0

Λ4QCD

f 2a

, (3.27)

where c0 = 1.46× 10−3, which leads to a minimum temperature ∼ 100 MeV for thevalidity of the fit.From Eq. (3.25), we see that the axion field begins to oscillate (realign) when

ma(Tosc) ≈ 3H(Tosc), (3.28)

where Tosc is the temperature of the Universe at the beginning of the oscillations.To obtain an expression for Tosc, we use the Friedmann equation in the radiation-dominated epoch, and the fit for the axion mass given by Eq. (3.26), which gives [100]

Tosc = 2.29 GeV(g∗(Tosc)

80

)(fa

1010 GeV

)− 2(4+n)

(ΛQCD

400 MeV

), (3.29)

where g∗(Tosc) is the number of relativistic degrees of freedom at temperature Tosc.Eq. (3.29) is valid for temperatures greater than 100 MeV, where Eq. (3.26) holds,and we have assumed a not too strong dependence on the temperature of g∗, which,for the range of interesting values of the axion decay constant in this work, 109 GeV <fa < 1011 GeV, varies between 80 and 87 [41], what would change the abundanceof axion dark matter by a factor of 1.03. If the axion mass changes slowly withtemperature, the number of axions with momentum zero per comoving volume isconserved [143], which leads to the conservation of energy, in the same volume, perunit mass of the axion in the oscillatory regime:

R3(t0)ρa,mis(t0)

ma(0)=R3(t1)ρa,mis(t1)

ma(T1), (3.30)

where t0, t1 are the cosmic times for T = 0 and T = T1, respectively, R is thescale factor of the Universe, and ρa,mis is the energy density of axions produced bythe misalignment mechanism. From Eq. (3.30), and the conservation of entropy percomoving volume, it is possible to obtain the dark matter abundance multiplied bythe square of the Hubble parameter in units of 100 km s−1Mpc−1 [100]

Ωa,mish2 ≡ ρa,mis(t0)h2

ρc,0= 4.75× 10−3

(fa

1010GeV

) 6+n4+n

, (3.31)

where we have set g∗(Tosc) = 80, and ΛQCD = 400 MeV.

50

3.3.2 Decay of global strings

Global strings are the first of topological defects that appear after the breaking ofthe global U(1)PQ symmetry, which happens when the PQ field φ acquires a VEV|〈φ〉| = vφ [144]. This process happens when T . vφ, and could be modeled by aHiggs potential [100]

V (φ) =λ

4(|φ|2 − v2

φ)2, (3.32)

where λ > 0 is a coupling constant. Nevertheless, in the 3-3-1 model, the symmetrybreaking potential is more complex, as can be seen in Eq. (3.11).Breaking of PQ symmetry leads to the formation of a densely knotted network ofcosmic axion strings, which oscillate under its own tension, losing their energy byradiation of axions [145]. The radiation process happens since the time of the PQtransition to the time of the QCD phase transition, and, considering the axions asmassless particles, the continuity equation gives the evolution of the energy densitiesof strings, ρstring, and axions produced by the string decays, ρa,string:

ρstring + 2Hρstring = −dρstring

dt

∣∣∣∣emission

, (3.33)

ρa,string + 4Hρa,string =dρstring

dt

∣∣∣∣emission

, (3.34)

where we have assumed equations of state of radiation and vacuum for axions andstrings, respectively [146]. Using results of numerical studies which provide the timedependence of ρstring, it is possible to obtain the abundance of radiated axions atthe present time [147],

Ωa,stringh2 = α×N2

DW ×(

fa1010GeV

) 6+n4+n

, (3.35)

where α = (7.3± 3.9)× 10−3, NDW is the number of domain walls in the model, andn = 6.68 is the same parameter that appears in Eq. (3.26).

3.3.3 Decay of string-wall systems

Domain walls appear as a consequence of breaking a discrete symmetry, which makesthe vacuum manifold to be made of several disconnected components. These domainwalls are attached by strings and occur at the boundaries between regions of spacewhere the value of the field φ is different. At this time, the temperature of theUniverse lies between the electroweak and QCD phase transition energy scales, andthis inhomogeneity of space-time is in contradiction to the assumptions of standardcosmology. So, it is necessary that these domain walls decay a certain time afterbeing formed [148]. The networks of domain walls bounded by strings begin tooscillate, and, when the tension of the walls is greater than the tension of the strings,their annihilation lead to the production of axions [149, 150].For our study, the number of domain walls attached to a single string is NDW = 3.

51

After the time when the surface mass density of domain walls dominates over thetension of the strings, the dynamics of the system is dominated by the tension ofdomain walls, where the energy density of the walls is described by a scaling solutionobtained numerically:

ρwall(t) =A(t)σwall

twith A(t) = Aform

(t

tform

)1−p, (3.36)

where tform is the time of wall formation, A the area parameter, and the value of pis obtained by numerical simulation [100]. We will refer to the cases p = 1 as theexact scaling, and p 6= 1 as the deviation from scaling.The energy density of walls can overclose the Universe, due to its dependence withthe inverse of the square of the scale factor, which decays at a slower rate thanthe corresponding to matter, ρ ∼ a−3, and radiation, ρ ∼ a−4. In our case, thisproblem is solved by the introduction of a Planck-suppressed operator in the effectivepotential for the axion field a, parametrized as in Eq. (3.19). In this case, the energydifference between the domain with lowest energy and its neighbor acts as a volumepressure on domain walls, which will collapse when this pressure becomes comparableto the domain wall tension (3.36), giving the decay time of domain walls [99]

tdec = Cd

[Aformσwall

tformΞv4φ(1− cos(2πN/NDW))

]1/p

tform, (3.37)

where

Ξ =|g|2N2

(vφMPl

)N−4

, (3.38)

is a dimensionless constant. Until this time, the long-lived domain walls will produceaxions with mean energy of the order of the axion mass.The current axion abundance is given by the expression [99, 100]:

Ωa,wallh2 =1.23× 10−6[7.22× 103]

32p β

(2p− 1

3− 2p

)[N4

DW

(1− cos

2πN

NDW

)]1− 32p

×(

Ξ

10−52

)1− 32p(

fa1010GeV

)4+3(4p−16−3n)

2p(4+n)

, (3.39)

where

β =C

( 32−p)

d A32p

form

ε3−2pa

,

is a parameter obtained from the results of numerical simulations, which, takinginto account the values reported in [100], is between 1.18 and 2.12.To conclude this section, we have seen that axions can be produced by three differentmechanisms, which leads to the result that the total abundance of axions in theUniverse can be written as the sum of all contributions, Eqs. (3.31), (3.35) and(3.39):

Ωah2 = Ωa,mish

2 + Ωa,stringh2 + Ωa,wallh

2. (3.40)

52

The total dark matter abundance due to axions is bound by the observational con-straint obtained by Planck collaboration [151]. In the next section, we will analyzethe behavior of each contribution to the total abundance, in order to establish asuitable region of parameters for the model analyzed in this work.

3.4 Observational constraints on the production ofaxion dark matter

In general, the total dark matter relic density due to axions in this 3− 3− 1 modeldepends on fa, g, NDW and ZN . The dependence on fa, g, andNDW is direct becauseΩa,mis, Ωa,string and Ωa,wall explicitly depend on these parameters [99].

Nevertheless, the dependence on ZN is indirect. Roughly speaking, this discretesymmetry constrains the order of the dominant gravity-induced operator gφN/MN−4

Pl .In other words, the discrete symmetry sets the exponent N which directly affects thetotal dark matter abundance due to axions. Actually, we have two discrete symme-tries, Z10 and Z11 (see Table 3.3), that stabilize the PQ mechanism, which impliesthat there are two cases to be considered, N = 10 and N = 11. On the other hand,the domain wall parameter, NDW, is set to be equal to 3 by the PQ symmetry andthe matter content in the model. Thus, we are interested to know if the model withZ10 or/and Z11 symmetry provides the total dark matter reported by the Planckcollaboration [151] when fa, g, take their allowed values, without conflicting withconstraints on axion phenomenology.

In order to do that it is convenient, first, to separately study the behavior ofthe three mechanisms for axion production, which is shown in Fig. 3.1. Specifically,the cyan and black lines show the axion abundances produced by misalignment andglobal string decay mechanisms, respectively. On the other hand, the blue linesshow the abundance of axion dark matter due to the decay of domain wall systemsfor N = 10 and N = 11, calculated for the coupling constant value |g| = 1. Twoshaded regions are also shown: the light red one corresponds to the exclusion regioncoming from the constraint of the over closure of the Universe [151], and the yellowregion gives the possible interval for the axion decay constant fa, for which noover abundance of axions from decay of global strings or domain walls is produced.Finally, the dark green line corresponds to the total abundance of axions, Ωah

2, asgiven by Eq. (3.40), obtained for the case N = 10 and |g| = 1. The case for N = 11is not shown because for all the considered values of fa the axion relic density isoverabundant.

From Fig. 3.1 some conclusions are straightforward. First, Ωa,mis and Ωa,string

grow when fa grows. Thus, in principle, these are dominant for the greater values offa (5.3×109 GeV . fa . 1.7×1010 GeV). However, the misalignment mechanism isalways sub-dominant because Ωa,string has an extra N2

DW = 9 global factor. Indeed,the misalignment mechanism contributes at most by ≈ 7% for the total dark matterdensity. In contrast, Ωa,wall is decreasing with fa and thus it dominates Ωa for thesmaller values of fa (3.6×109 GeV . fa . 5.3×109 GeV). That can be understoodrealizing that the domain-wall time decay is larger for smaller fa values, making

53

the domain walls more stable and, in this way, explaining why this mechanism con-tributes more for the axion relic density when fa is smaller. The opposite behaviorof Ωa,string and Ωa,wall allow to set an upper and lower bound on fa. For |g| = 1,fa is constrained to be 3.6 × 109 GeV < fa < 1.7 × 1010 GeV in order to satisfyΩa,wallh

2 and Ωa,stringh2 . ΩPlanck

DM h2 [151]. Actually, the interval of allowed fa valuesis slightly thinner because all of the three axion production mechanisms contributesimultaneously. Also, note that the fa upper bound above is independent on thevalue of N and on the value of |g|, as can be seen from Eq. (3.35). In contrast,the lower bound is only valid for the case of N = 10. Actually, the case of Z11 iscompletely ruled out and, for this reason, our analysis will be concerned exclusivelywith the Z10 symmetry case.

0

0.05

0.1

0.15

0.2

9 10 11

Ωah

2

log10 fa(GeV)

0

0.05

0.1

0.15

0.2

9 10 11

N = 10 N = 11

Ωa,stringh2

Ωa,wallh2 Ωa,wallh

2

Ωa,mish2

Ωah2>ΩPlanckDM h2

Figure 3.1: Relic density of non-thermal axion dark matter in the 3− 3− 1 model,assuming exact scaling, p = 1, and |g| = 1. The central values of the parameters inEqs. (3.35) and (3.39) together with NDW = 3 have been used. The vertical dashedlines limit regions with over production of axions by decay of domain walls (left line)and strings (right line), while the horizontal red line is the experimental constraintΩah

2 = ΩPlanckDM h2, the cyan and black lines show the axion abundances produced by

misalignment and global string decay mechanisms, respectively. Finally, the bluelines show the abundance of axion dark matter due to the decay of domain wallsystems for N = 10 and N = 11, calculated for the coupling constant value |g| = 1.

Once we have gained a general knowledge about the behavior of Ωah2 as a func-

tion of fa for |g| = 1, we can go further studying the parameter space for the Z10

case, allowed by the axion phenomenology.

54

In particular, in Fig. 3.2 we show the parameter space fa − |g| for the casesof exact scaling (p = 1, left frame) and deviation from scaling (p = 0.926, rightframe), where the range of values of the coupling constant g has been chosen suchthat |g| ≤

√4π. The blue curves correspond to regions where the total axion dark

matter abundance is equal to ΩPlanckDM h2, taking into account the uncertainties in the

parameters α and β in Eqs. (3.35) and (3.39). Notice that for a given value of fa,|g| is bounded by these lines. Larger values of |g| imply Ωah

2 < ΩPlanckDM h2. The light

blue shaded region is ruled out by the over closure of the Universe for the case ofthe parameter β = 2.12 in Eq. (3.39) and for the α = 7.3− 3.9 = 3.4 factor in Eq.(3.35).

From the remaining region, it is possible to exclude another large part applyingthe axion mass stability condition, ma,QCD > ma,gravity (see the discussion near Eq.(3.23)). Because ma, gravity is directly proportional to |g| and fN−2

a , cf. Eq. (3.23),and m2

a, QCD is inversely proportional to f 2a , cf. Eq. (3.24), the forbidden region,

denoted by the light red color, is in the top right part of the fa − |g| plane.In addition, Fig. 3.2 shows three dark red lines which correspond to the NEDM

constraint, given by Eq. (3.22), for different values of δD. It is important to realizethat δD values of order one (not shown) do not give allowed regions in the parameterspace. It is necessary to allow δD . 10−5 in order to have non-excluded regions whichare below the lines.

In particular, we calculate the maximum values of δD that give allowed regionsin the parameter space. The corresponding results, in the cases of exact scaling(p = 1) and deviation from scaling (p = 0.926), are

δD =

(0.4− 4.1)× 10−5 Exact scaling,(2.9− 9.5)× 10−6 Deviation from scaling,

(3.41)

which have been obtained by taking |g| =√

4π, and considering the uncertaintiesin the parameters of the three axion production mechanisms. Lower values of |g|would require higher tuning on the δD parameter, with values of the order 10−8 asshown in Fig. 3.2. In general, for |g| fixed, the tuning on δD depends on the decayconstant fa and the mechanism of axion dark matter production: if the decay ofdomain walls was dominant (left side of the curves), the tuning would be less severethan if the production by string decay (right side of the curves) was the dominantone.

Also, in Fig. 3.2 is shown that for a δD small enough in order to satisfy theNEDM condition, and for a given |g| value between 5 × 10−2 and

√4π, there are

two separated regions for fa where axions can make up the total DM relic density.For instance, taking |g| =

√4π and considering the uncertainties in the parameters,

these regions and their corresponding axion masses for the exact scaling case, are:

fa ≈

(2.8− 3.5)× 109 GeV −→ ma ≈ (1.7− 2.1)× 10−3 eV

(1.1− 1.2)× 1010 GeV −→ ma ≈ (5− 5.4)× 10−4 eV.(3.42)

In the first range for fa, the production of dark matter is mainly through the decayof domain walls, while in the second range it is due to the decay of strings. Taking

55

-6

-5

-4

-3

-2

-1

0

10 11

log 1

0|g

|

log10 fa(GeV)

N = 10 Exact scaling

-6

-5

-4

-3

-2

-1

0

10 11

δD

=10 −

6

δD

=10 −

8

δD

=10 −

10

Ωah2 > ΩPlanckDM h2

(a)

-6

-5

-4

-3

-2

-1

0

10 11

log 1

0|g

|

log10 fa(GeV)

N = 10 Deviation from scaling

-6

-5

-4

-3

-2

-1

0

10 11

δD

=10 −

6

δD

=10 −

8

δD

=10 −

10

Ωah2 > ΩPlanckDM h2

(b)

Figure 3.2: Observational constraints on the parameter space fa−|g| in the 3−3−1model, assuming exact scaling (a) and deviation from scaling (b). These plots corre-spond to the Z10 discrete symmetry, and NDW = 3. The shaded regions in light redand light blue correspond to regions of the parameter space where the constraintsgiven by ma,QCD > ma,gravity and Ωah

2 6 ΩPlanckDM h2 are violated, respectively. More-

over, the regions above the straight red lines correspond to excluded regions set bythe NEDM condition, as given by Eqs. (3.21) and (3.22), for three different choicesof the δD parameter.

smaller values for |g|, will lead to more stringent intervals for both fa and ma. Forthe case of deviation from scaling, we find fa ≈ (3.4−3.6)×109 GeV, correspondingtoma ≈ (1.7−1.8)×10−3 eV, when the domain walls decay is the leading productionmechanism, and fa ≈ (1.1− 1.2)× 1010 GeV, leading to ma ≈ (5− 5.4)× 10−4 eV,for the string decay as the dominant contribution.

Finally, for values of |g| of order one, we can make predictions regarding theobservability of axion in current and/or future experiments. Specifically, the axioncoupling to photons, gaγγ, depends on the fa decay constant as [152]

gaγγ =α

2πfaCaγγ =

α

2πfa

[Caγγ −

2

3

4 + z

1 + z

], (3.43)

where α is the fine strucure constant, z = mu/md [153], and

Caγγ =1

NDW

∑

f

XfQ2f , (3.44)

with Xf and Qf are the UPQ and electric charge of the fermion f , respectively. Thelast term in (3.43) appears only when the light quarks carry a UPQ charge [127].

Taking into account that the electric charge operator is given by Q = 12(T3 −

1√3T8) + X, with X the charge under U(1)X and Ta the SU(3) generators, we can

56

use equations (3.1) and the U(1)PQ charges of the model, cf. Table (3.2), in orderto obtain Caγγ = Caγγ = −4/3, in this case.

With this information, in Fig 3.3 we plot gaγγ as a function of ma for the regionswhere axions make up the total dark matter relic density and for two different valuesof |g|, specifically |g| = 0.1 and |g| = 1. This figure clearly shows two allowed regionsfor |g| = 0.1: ma ≈ (0.4− 0.6)× 10−3 eV with gaγγ ≈ (4.5− 5.9)× 10−13 GeV−1 andma ≈ (0.9− 1.3)× 10−3 eV with gaγγ ≈ (1.1− 1.6)× 10−12 GeV−1, and one regionfor |g| = 1: ma ≈ (1.4− 1.8)× 10−3 eV with gaγγ ≈ (1.8− 2.2)× 10−12 GeV−1. Thereason why there is only one region for larger |g| values is that the gravitational massgrows with |g| and thus, it conflicts with the conditionma, QCD ma, gravity for loweraxion masses. Moreover, it is notable that for the range with larger masses (blueline), the axion parameters of this 3 − 3 − 1 model are very close to the projectedregion which is going to be explored by the IAXO experiment [99, 102].

Axion

CDM

IAXO

ALPS-II

Ha

losc

op

es

AD

MX

g=1

g=0.1

g=0.1

-6 -5 -4 -3 -2-14

-13

-12

-11

Log10 ma [eV]

Lo

g1

0|g

aγγ|[

Ge

V-

1]

Figure 3.3: Projected sensitivities of different experiments in the search for axiondark matter. The green regions show sensitivities of light-shining-through-wall ex-periments like ALPS-II [103], of the helioscope IAXO [102], of the haloscopes ADMXand ADMX-HF [104, 105]. The yellow band corresponds to the generic predictionfor axion models in QCD. In addition, the two (one) thick red (blue) lines standfor the predicted mass ranges and coupling to photons in this model, for |g| = 0.1(|g| = 1), where axions make up the total DM relic density.

57

3.5 Models with different discrete symmetries andnumber of domain walls

As studied in previous sections, the production of axions is given by three processes:misalignment, decay of global strings and decay of string-wall systems. The first ofthese is a subdominant channel and depends only on the symmetry breaking scale,of energy fa.On the other hand, the other two processes in addition to depending on fa, changewith the number of domain walls NDW , an amount that is related with the particlecontents of the model and the UPQ charge assignment in the 3− 3− 1 model.

Furthermore, for the decay of the string-wall system the discrete symmetry ZD,used to stabilize the axion mass in the model, can change dramatically the parameterspace, as shown in the last section.

In this section we are going to compare models that differ in NDW and ZD,showing the impact of these quantities over the observational constraints previouslyanalyzed.

The number of domain walls NDW , is defined as the ratio between the periodicityof the axion field a and 2πfa [154]

NDW =periodicity of a

2πfa, (3.45)

where fa is the decay constant of the axion. When NDW > 1, the theory has adiscrete symmetry ZN that is spontaneously broken when the a field gets a vacuumexpectation value, in such a way that the vacuum of the theory is degenerate andNDW domain walls are formed, where [127]

NDW = 2

∣∣∣∣∣∑

f=L

Tr XfT2a (f)−

∑

f=R

Tr XfT2a (f)

∣∣∣∣∣ , (3.46)

where Xf is the U(1)PQ charge of the quark f , and Ta are the generators of SU(3)C ,normalized such that T 2

a = 12.

In order to show how the number of domain walls and ZD can modify the allowedparameter space of a model, we compare three different 3−3−1 models [14, 127, 155],where the particle contents, the UPQ charge assignment and the ZD symmetry aredifferent.

Now, we present the most relevant aspects of these models, classified dependingon the number of domain walls NDW and the discrete symmetry ZD used to stabilizethe axion:

• NDW=1, Z13, introduced in ref. [127], where the quark sector is described by

QmL = (dm, um, jm)TL ∼ (3, 3,−1/3), m = 1, 2

Q3L = (d3, u3, j3)TL ∼ (3, 3, 2/3),

uαR ∼ (3, 1, 2/3), dαR ∼ (3, 1,−1/3), α = 1, 2, 3

JR ∼ (3, 1, 5/3), jmR ∼ (3, 1,−4/3) (3.47)

58

and the assignment of UPQ charges is such that Xd = −Xu and Xj = −XJ ,but is assumed that Xd = 0, in such way that the only contribution to NDW

is given by Xj.

• NDW=2, Z9, presented in ref. [155], where the colored and charge carryingparticles in this model are

QaL = (da,−ua, Da)TL ∼ (3, 3, 0), a = 1, 2

Q3L = (u3, d3, U)TL ∼ (3, 3, 1/3),

u′nR = (uiR, UR) ∼ (3, 1, 2/3),

d′mR = (diR, DaR) ∼ (3, 1,−1/3), (3.48)

where i = 1, 2, 3 and n,m vary from 1 to 4 and 1 to 5, respectively. Thefull assignment of UPQ charge is given in table II in [155], from which wehave extracted the values of XPQ for colored particles, useful to determine denumber of domain walls in the model, using eq. (3.46).

Table 3.4: UPQ charge of the color-charged particles.Table adapted of [155].

Field QaL Q3L uaR u3R UR djR DaR

XPQ 0 -4 -3 -4 -7 -1 3

• NDW=9, Z10, presented in [14], and which corresponds to the model analyzedsection 3.2.

In figure 3.4, a comparison of the predictions for the axion abundance for themodels is shown, when the coupling parameter of the high dimensional operatorgiving a gravitational mass to the axion is taken with |g| = 1. The upper panelgives the results for the model in ref. [127], the middle panel for the model in [155],and the lower panel shows the results for [14]. Each panel shows the order of thediscrete symmetry ZN and the corresponding number of domain walls NDW, and thecontributions of the three mechanisms for the production of axion DM, as indicatedby the dashed lines. The blue bands in each panel give the total axion abundance,which can be compared with the dark matter abundance measured by [13], givenby the horizontal red line. The shadowed region in yellow corresponds to valuesof the axion decay constant fa for which the gravitational mass induced by highdimensional operators is greater than the QCD axion mass given by Eq. (3.24)2.From this figure, we can see that the model with NDW = 1 has only one range ofvalues of fa for which the total abundance of dark matter can be explained by themodel, fa = (4.2 − 6.6) × 1010 GeV, in contrast to the models with NDW 6= 1 thathave two possible regions of fa where the model could explain all dark matter in theuniverse.On the other hand, due to the restriction over the axion mass,ma, QCD < ma, grav, for

2This region is absent in the plot for NDW = 1, because the high order of the discrete symmetryrequires a minimum value of fa = 2.2 × 1012 GeV, well outside the region where the model canexplain the total DM abundance.

59

the case NDW = 2 the region with upper values of fa giving the total DM abundanceis ruled out, and, in the allowed region, axion production is dominated by the decayof domain walls.Finally, for the model withNDW = 3, there are allowed regions with axion productionby strings and domain walls, but it is important to realize that for the greatest favalues the gravitational mass is less than the QCD mass, but they are comparable,as we will see later.

In figure 3.5, we show the allowed region for |g| as a function of fa, for the modelswith NDW = 2 (upper panel) and NDW = 3 (lower panel). As the DM abundancefrom the decay of string-wall systems is independent of |g| for the model with NDW =1, this model has no constraint on this parameter. The inner, central and outerblue lines in the figure where obtained taking the minimum, central and maximumvalues of the DM abundance reported by [13], and the blue region corresponds topoints where the axion abundance is greater than that value. The yellow region isexcluded due to the constraint ma, QCD > ma, grav. The green lines correspond tothe constraint given by Eq. (3.22), where the allowed parameter space is below thelines, characterized by the phase of the coupling g, as indicated.

From the NEDM condition, it is clear that the maximimum value of δD requiredto have an allowed region of the parameter space is greater for the case NDW = 2than for the model with NDW = 3, meaning that the first model needs less finetunning over this parameter. On the other hand, we can see that the minimumvalue allowed for |g| is of the order 10−13 for the first model, and 3 × 10−5 for thesecond.

An important characteristic of the model with NDW = 2 is that, in order toexplain all DM in the universe for greater values of fa it is necessary to imposesmaller values of g, in contrast with the model with NDW = 3 which has two possibleranges of fa where the same value of |g| can work. Nevertheless, for greater valuesof fa, the axion mass could have a gravitational contribution comparable with theQCD mass.

Finally, in the experimental prospects shown in figure 3.6, we present the resultson the axion coupling to photons |gaγγ|, obtained from the intervals of fa for whichthe total DM abundance can be in the form of axions, as obtained from figure 3.5(in this case we have shown the total contribution, in contrast with Fig. (3.3),that shows only the linear regime). From this figure, we can see that the model withNDW = 2 is close to be tested with the measurements of the IAXO experiment, whilethe MADMAX projected experiment could be able to detect axions in the modelwith NDW = 1. Finally, for the model with NDW = 3, the intervals with lower axionmass fall out of the lines for which the total axion mass comes from QCD effects(labelled by the values of E/NC, where E and NC are anomaly coefficients), due tothe contribution of ma, grav, which is still allowed by all constraints considered.

3.6 ConclusionsIn this work, we consider a version of an alternative electroweak model based onthe SU(3)L ⊗ U(1)X gauge symmetry, the so called 3 − 3 − 1 models, when the

60

color gauge group is added. For this version, which includes right-handed neutrinos,it is shown in Ref. [95] that the PQ mechanism for the solution of the strong CPproblem can be implemented. In this implementation, the axion, the pseudo Nambu-Goldstone boson that emerges from the PQ-symmetry breaking, is made invisibleby the introduction of the scalar singlet φ ∼ (1, 1, 1) whose VEV, vφ ≈ fa, is muchlarger than vSM, and any other VEV in the model. Moreover, the axion is alsoprotected against gravitational effects, that could destabilize its mass, by a discreteZN symmetry, with N = 10, 11.

Once we have set this consistent scenario, we investigate the capabilities of thisaxion, produced in the framework of this particular 3 − 3 − 1 model, to be a post-inflationary cold dark matter candidate. We started focusing in the axion-productionmechanisms. As it was explained, from Fig. 3.1 we see that the vacuum misalignmentmechanism does not dominate the DM relic abundance and, if it were the onlyproduction mechanism in action, an upper bound for fa could be set by imposingthat it should account for all the DM abundance, i.e., Ωa,mish

2 = ΩPlanckDM h2, and

we would find the corresponding value fa ≈ 1.5 × 1011 GeV, for the parametersdetermined by the model, in this case NDW = 3.

However, there are two other more efficient mechanisms due to the decay oftopological defects: cosmic strings and domain walls. As the curves for Ωa,stringh

2

and Ωa,wallh2 grow in opposite directions, relatively to the fa values, we can determine

an upper bound and a lower bound for fa by imposing the total Ωah2 matches the

observed Planck results. This is the case when we add up all the contributions forN = 10, and we find 3.6× 109 GeV < fa < 1.7× 1010 GeV. However, we would liketo stress that this is not the case for N = 11. For N = 11 there is no value of fafor which the addition of the partial abundances lies below the observed result. Itmeans that Z11, which possesses the desirable characteristic of stabilizing the axion,is not appropriate for the axion-production issue since it makes the domain wallmechanism too efficient and overpopulates the Universe.

As it can be seen from Fig. 3.1, for any fixed allowed value of |g|, there are twovalues of fa that are in agreement with the value of ΩPlanck

DM h2. In fact they are regions,if we take into account the uncertainties following the discussion of Fig. 3.2. Outsidethese regions, the axion abundance will be a fraction of ΩPlanck

DM h2. See the solid darkgreen curve in Fig. 3.1 for |g| = 1. If this happens to be the case, i.e., if thesepredicted regions are somehow excluded by future experimental data for the axionmass value, another kind of DM will be needed. We have also found special valuesfor δD, (0.4 − 4.1) × 10−5, by requiring the minimal compatible intersection regionbetween the curves that obey the NEDM and ΩPlanck

DM h2 constraints. This value wasobtained considering the maximum value of |g|, i.e., |g| =

√4π, cf. Fig. 3.2a. For

lower values of |g|, higher tuning on δD is required. However, it seems unnatural torequire severe levels of tuning on δD, since for this quantity a tiny value is the resultof the difference between two terms that have completely different origins.

Regarding the capabilities of detecting axion dark matter in this model, Fig. 3.3shows the sensitivities of several experiments in the ma−gaγγ plane. In this plot, thethick blue and red lines are the regions where the axion abundance is responsible forall the observed DM. These lines were obtained by using |g| of order one. Moreover,

61

the blue region corresponding to masses of order meV and gaγγ ≈ 10−12 GeV−1, liesvery close to the projected IAXO sensitivity, so that it will be reachable in the nearfuture.

Finally, we have made a comparison of the predictions on axion DM in differentversions of the 3− 3− 1 model (Figures 3.4,3.5 and 3.6), considering the constraintscoming from DM relic abundance, the measurements on the NEDM and the dom-inance of the mass coming from instanton effects over the contribution of higherdimensional operators. We have seen that the three models have a wide allowedparameter space, and analised the possible prospects of observation.

62

-2

-1

-4

-3

-2

-1

-2

-1

8 9 10 11

Misalignment

Strings

Walls

-2

-1

N

DW

= 1

N = 13

log10Ω

ah2

-4

-3

-2

-1

N = 9

N

DW

= 2mgrav

a > mQCD

a

log10 fa(GeV)

-2

-1

8 9 10 11

N = 10

N = 9

N

DW

= 3

N

DW

grav QCD

Figure 3.4: Axion abundance in the 3− 3− 1 models considered in this work, char-acterized by different orders of the discrete symmetry stabilizing the axion solutionto the strong CP problem and numbers of domain walls NDW. The blue bands ineach panel give the total axion abundance, which can be compared with the darkmatter abundance measured by [13], given by the horizontal red line. The shadowedregion in yellow corresponds to values of the axion decay constant fa for which thegravitational mass induced by high dimensional operators is greater than the QCDaxion mass.

63

-14

-12

-10

-8

-6

-4

-2

0

8 9 10 11

δD=10 −

6

δD=10 −

10

-5

-4

-3

-2

-1

0

9 10 11

log10|g|

log10 fa(GeV)

-14

-12

-10

-8

-6

-4

-2

0

8 9 10 11

N = 9NDW = 2

Ωah2 > ΩPlanckh

2

mgrav

a

>m

QCD

a

log10|g|

log10 fa(GeV)

-5

-4

-3

-2

-1

0

9 10 11

N = 10NDW = 3

δD=

10 −6

δD=

10 −10

Figure 3.5: Constraints on the coupling g of gravitationally induced operators andthe axion decay constant fa, for the models with NDW = 2 (up) and NDW = 3(down). The blue region corresponds to points where the axion abundance is greaterthan the DM abundance reported by [13]. The yellow region is excluded due to theconstraint ma, QCD > ma, grav. The green lines correspond to the constraint givenby Eq. (3.22), where the allowed parameter space is below the lines.

64

AxionCDM

IAXO

ALPS-II

Haloscopes

ADMX

(2,1)

(2,0.1)

-)

(3,1)

(3,0.1)

E/NC=2/3

E/NC=-4/3

-6 -5 -4 -3 -2-14

-13

-12

-11

Log10 ma [eV]

Log10|g

aγγ|[GeV-1]

Figure 3.6: Sensitivity to the axion-photon coupling gaγγ for different experiments,adapted from [156]. The small colored lines show the ranges of the axion mass forwhich the total DM of the universe can be made of axions, where the numbers inbrackets alongside each line represent the number of domain walls in the model,NDW, and the assumed value of the |g| coupling. [15]

65

Chapter 4

Conclusions

We have studied two important aspects of the physics of the early universe: the infla-tionary epoch, which solves important problems of the standard cosmological model(as the flatness and horizon problems), and the dark matter puzzle, responsible forthe formation of the structures we see in the universe today.

The analysis of an inflationary era in the universe has been performed in a mod-ified theory of general relativity, where higher derivative terms have been added tothe Einstein-Hilbert action. The analysis of these contributions in terms of smallperturbations of the Starobinsky model of inflation has led us to find that thesehigher derivative terms can describe successfully an inflationary period in the earlyuniverse, and the cosmological observables obtained within its framework are con-sistent with the current constraints on these quantities, for suitable values of theperturbative parameter.

On the other hand, we have studied dark matter in the form of weakly interactingslim particles (WISPs) in the framework of the SU(3)c ⊗ SU(3)L ⊗U(1)X with theaddition of a scalar singlet, which solves the strong CP problem and introducesan axion particle that can be identified with dark matter. We have reviewed themechanisms for the production of axions in the early universe, which turn out to beof a non-thermal character, as opposed to the usual processes leading to the weaklyinteracting massive particles (WIMPs), produced during thermal equilibrium in theearly universe.

We have analyzed the implementation of a gravity stable Peccei-Quinn mecha-nism in a specific realization of the model, analyzed constraints coming from thestability of the axion mass, the neutron electric dipole moment and the dark mat-ter relic abundance. For the same realization of the model, we have studied theobservational prospects.

Finally, taking into account that there are different realizations of SU(3)c ⊗SU(3)L ⊗ U(1)X including axions in their physical spectrum, characterized by dif-ferent stabilizing symmetries and different PQ charge assignments (which determinethe number of domain walls for each model), we performed a comparison of the darkmatter predictions of three different models and analyzed the constraints on eachmodel coming from the previously mentioned observables.

66

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