Instituto de Física “Gleb Wataghin”
Ana Rubiela Romero Castellanos
Higher derivative corrections to the R
+ R
2inflationary model and axions in the
3
− 3 − 1 model
Correções de derivadas superiores ao modelo
inflacionário R + R
2e áxions no modelo 3
− 3 − 1
CAMPINAS
2019
Higher derivative corrections to the R
+ R
2inflationary model and axions in the
3
− 3 − 1 model
Correções de derivadas superiores ao modelo
inflacionário R + R
2e 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 requirements for the degree of Doctor in Sciences.
Tese apresentada ao Instituto de Física “Gleb Wataghin” da Universidade Estadual de Campinas como parte dos requisitos exi-gidos para a obtenção do título de Doutora em Ciências.
Orientador: Prof. Dr. Pedro Cunha de Holanda Este exemplar corresponde à versão final
da tese defendida pela aluna Ana Rubiela Romero Castellanos e orientada pelo Prof. Dr. Pedro Cunha de Holanda.
CAMPINAS
2019
Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174
Romero Castellanos, Ana Rubiela,
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 3-3-1
Palavras-chave em inglês: Starobinsky inflation
Higher-derivative corrections Slow-roll approximation Dark matter (Astronomy) Axions
Área de concentração: Física Titulação: Doutora em Ciências Banca examinadora:
Pedro Cunha de Holanda [Orientador] Orlando Luis Goulart Peres
Donato Giorgio Torrieri Alex Gomes Dias Celso Chikahiro Nishi
Data de defesa: 18-09-2019
Programa 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
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
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 to aiding 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 trusted me, taught me that surrendering is not an option and set up the opportunity to work with her and with great physicists, as professor Ilya Shapiro from UFJF, who during my stay in UFJF was a great tutor and advisor, and taught me a lot of things. I have to say that it was a huge privilege to have the opportunity to work with one of the creators of the inflationary theory, professor Alexei A. Starobinsky, whom I want to thank for his guidance and all his comments and suggestions during several stages 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 never felt 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 de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
Nesta tese, analisaram-se dois tópicos: o modelo inflacionário de Starobinsky com um termo de derivada superior como uma pequena perturbação, e a inclusão de áxions num modelo SU(3)c⊗SU(3)L⊗U(1)X (3−3−1) como um possível candidato para 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 época inflacionária 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 observacionais atuais sob essas quantidades.
Na segunda parte desse trabalho, foi obtida a abundância de matéria escura na forma de áxions no contexto de um modelo 3 − 3 − 1, levando em conta suas contribuições do mecanismo de desalinhamento, e devido ao decaimento de defeitos topológicos, 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 matéria escura. Finalmente, foi feita uma comparação de três versões diferentes de modelo 3− 3 − 1 com áxions em seu espectro físico, mostrando as diferenças, vantagens e desvantagens de cada modelo, quando os vínculos atuais sob a abundância relíquia de 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.
In this thesis, we address two topics: the Starobinsky inflationary model with a higher derivative term as a small perturbation, and the inclusion of axions in a SU (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 model as a small perturbation, and it is shown that it is possible to have an inflationary epoch in the early universe and, at the same time, give values for the spectral index and the scalar-to-tensor ratio which are consistent with the current observational constraints on this quantities.
In the second part of this work, we obtained the abundance of axion dark matter in the context of a 3 − 3 − 1 model, taking into account its contributions from the misalignment mechanism, and due to the decay of topological defects, domain walls and string-walls systems. The results were compared with current observations, and we found the allowed parameter space, showing that the axion in this scenario can be an appealing candidate to dark matter. Finally, we made a comparison of three different 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 on the relic abundance of dark matter and prospects of axion detection are taken into account.
Keywords: Starobinsky inflation, higher-derivative corrections, slow-roll approxi-mation, dark matter, axions, 3 − 3 − 1 model.
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 constraints imposed by Planck data [13], as will be shown later. . . 33 2.2 RR correction to the Starobinsky model of inflation for different
values of the parameter k. The black continuous line corresponds to the original R + αR2 model, and the colored dashed lines to nonzero values of k. . . 34 2.3 The slow roll parameters (up) and η (down) for the model with
RR term as functions of the inflaton field χ. . . 36 2.4 Observational constraints set by the Planck collaboration [13, 71] on
the scalar-to-tensor ratio r and scalar spectral index ns, and the pre-diction for these quantities in the R + R2+ RR model. The allowed regions for this model are presented in red and green color for nega-tive and posinega-tive values of the k parameter, respecnega-tively. This figure is an original result of this thesis and appears in [12]. . . 38 2.5 Observational constraints on the slow roll parameters set by Planck
collaboration [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 of the scalar potential. This figure is an original 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 the parameters in Eqs. (3.35) and (3.39) together with NDW = 3 have been 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 Ωah2 = 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 of axion dark matter due to the decay of domain wall systems for N = 10 and N = 11, calculated for the coupling constant value |g| = 1. . . 53
These plots correspond to the Z10 discrete symmetry, and NDW = 3. The shaded regions in light red and light blue correspond to regions of the parameter space where the constraints given by ma,QCD >
ma,gravity and Ωah2 6 ΩPlanckDM h2 are violated, respectively. Moreover,
the regions above the straight red lines correspond to excluded regions set by the NEDM condition, as given by Eqs. (3.21) and (3.22), for three different choices of the δD parameter. . . 55 3.3 Projected sensitivities of different experiments in the search for axion
dark 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 band corresponds to the generic prediction for axion models in QCD. In addition, the two (one) thick red (blue) lines stand for 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. . . 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 the axion solution to the strong CP problem and numbers of domain walls NDW. 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 yellow corresponds to values of the axion decay constant fa for which the gravitational mass induced by high dimensional operators is greater than the QCD axion mass. . . 62 3.5 Constraints on the coupling g of gravitationally induced operators
and the 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 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 of the axion mass for which the total DM of the universe can be made of axions, where the numbers in brackets alongside each line represent the number of domain walls in the model, NDW, and the assumed value of the |g| coupling. [15] . . . 64
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 such
that 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 in
the considered 3 − 3 − 1 model. . . 47 3.4 UP Q charge of the color-charged particles.Table adapted of [155]. . . . 58
Introduction 13
I
R + R
2inflationary model
16
1 Inflationary cosmological standard model 17
1.1 The standard big-bang cosmology and motivations for the inflationary
paradigm . . . 17
1.2 Inflationary model . . . 19
1.2.1 Slow-roll approximation . . . 21
1.2.2 Cosmological perturbations in inflation . . . 22
2 On higher derivative corrections to the R + R2 inflationary model 24 2.1 The R + R2 Model . . . 25
2.2 Treating RR term as a new degree of freedom . . . 26
2.2.1 Weak field approximation . . . 28
2.2.2 Simplifying Ansatz R = r1R + r2 . . . 29
2.3 Treating RR term as a small perturbation to the R + R2 model . . 30
2.3.1 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 42 3.1 A brief review of the model . . . 433.2 Implementing a gravity stable PQ mechanism in the model . . . 45
3.3 The Production of Axion Dark Matter . . . 48
3.3.1 Misalignment mechanism . . . 48
3.3.2 Decay of global strings . . . 50
3.3.3 Decay of string-wall systems . . . 50
3.4 Observational constraints on the production of axion dark matter . . 52
3.5 Models with different discrete symmetries and number of domain walls 57 3.6 Conclusions . . . 59
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 of density and temperature.
But how the universe could evolve from such conditions to be what we observe today, that is, planets, stars, galaxies, clusters of galaxies, etc? This is a question that can not yet be answered conclusively.
The observable universe can be described by the standard cosmology (SC), a model that has seven parameters [1]. Five of them determine the background of the universe: H0, ΩB, ΩDM, ΩΛ, ΩR, where H0 is the Hubble parameter measured today, and Ωi = ρi/ρc with i = B, DM, Λ, R, representing the abundances of baryons, dark matter, dark energy and radiation, respectively, calculated as the ratio of the density of the corresponding component (ρi) and ρc, the energy density that makes the universe to be flat, called the critical density. The two remaining parameters, the amplitude As and the spectral index ns characterize the spectrum of the initial fluctuations, the seeds for the structures that we see today in the universe, and that should be printed on the cosmic microwave background (CMB). Nevertheless, anything can be said about the origin of these fluctuations inside the SC.
The inflationary paradigm claims that around ∼ 10−40 to 10−32 seconds after the big-bang, the universe developed an inflationary era in which the scale factor of the universe increased in an accelerated way. It initially emerged as a mechanism to answer some questions like why the universe is so flat or why the universe is homogeneous, among others, but soon after its proposal it was realized that in this kind of scenarios the origin of primordial fluctuations can be explained and is related with 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 early universe, which can be discriminated by their predictions for the spectral index [2]. The simplest model uses a scalar field, called inflaton, with a scalar potential sufficiently flat in order to disregard the kinetic energy of the field and assume that the 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 its analysis is more involved [3]. It is important to remark here that whatever the number of fields in the models, they always require the addition of new fields in the theory, not included in the standard model of particle physics (SM).
in the analysis of corrections to the gravitational action, including terms involving higher powers of the curvature tensor, which is a natural way to obtain inflationary scenarios, without the previous imposition of the existence of the inflaton field or any number of new fields [4].
The model R + R2, first proposed by Starobinsky [5], is the simplest extension of the Einstein-Hilbert Lagrangian, and has the characteristic that the equations of motion include fourth-order derivatives of the metric instead of only second-order ones [6], and provided an inflationary solution even when the cosmological constant Λ is zero [7]. One feature that makes this kind of model interesting is the fact that it 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 Starobinsky model, sixth-order derivatives of the metric appear in the equations of motion, and an inflationary era in the universe can be developed. This model is conformally equivalent 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 treat the extra RR term as a small perturbation to the Starobinsky model, and show that this theory is consistent with observations and also with some previous results obtained when the theory is mapped to a model with two scalar fields [4,9,10], and with 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 with Flavia Sobreira, Ilya L. Shapiro and Alexei A. Starobinsky. This work opens the possibility to treat other higher derivative terms as perturbations of the Starobinsky model and not of the Einstein-Hilbert action during the inflationary era, which is quite 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 also interested in the abundance of dark matter, measured in terms of the cosmological parameter ΩDM. The last reports of the Planck collaboration [13] show that around 95%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 universe are in the form of dark matter and dark energy, respectively, and just 5% is made of baryonic matter, that can be described by the SM.
In order to describe this unknown matter in the universe, numerous candidates have been proposed: weakly interacting massive particles (WIMPs), axions, Kaluza-Klein particles, massive astrophysical compact halo objects (MACHOs),..., but any attempt to use a new particle to explain the features of DM needs models beyond the SM. Within the large number of candidates for DM, axions look like a compelling one due to the fact that they can solve the strong CP problem and also explain the abundance of dark matter in the universe, evading the current experimental constraints on these particles.
In the second part of this thesis, we work in the context of models with a larger symmetry group than the one of the SM, specifically with the model based on SU (3)c⊗ SU(3)L ⊗ U(1)x gauge group, called the 3 − 3 − 1 model, in order to include an axion in its physical spectrum, analyze the different mechanisms for its
production during the early universe, giving a relic abundance consistent with the current constraints, and determine its prospects of detection.
The 3 − 3 − 1 models are appealing extensions of the SM, because these kind of models have several interesting characteristics. For example, the number of fermion generations 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 the total abundance of dark matter in the universe. On the other hand, in these scenarios neutrinos are massive particles and, it has been shown that new scalars in the model can contribute to the muon anomalous magnetic moment.
We analyze the scenario where the Peccei-Quinn (PQ) symmetry is broken after the inflationary era. In such a case, in addition to the misalignment mechanism for the production of axions, the decay of topological defects (domain walls and string-walls systems) also contributes to the abundance of axion dark matter. By first time we obtain the total abundance for axions in the case of a 3 − 3 − 1 model and compare it with current observations. The results of these analyses appear in the paper [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 symmetry used to stabilize the axion mass against gravitational effects is strongly constrained by observations on the DM relic abundance, and depending on the characteristics of 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 including axions 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 different versions of the 3 − 3 − 1 model with axions was made, analyzing how the number of domain walls and the particle content of the models can modify the axion DM abundance of each model. Finally, we analyze both the QCD and gravitational contribution to the axion mass, and not just the linear approximation, as done in previous works involving axions in other models. The main results of these analyses were published in [15], written in collaboration with C. E. Alvarez-Salazar and B. L. Sánchez-Vega.
R + R
2
inflationary model
Chapter 1
Inflationary cosmological standard
model
Current observations point to the establishment of the Standard Cosmological Model (SC) as a suitable description of the universe on large scales. The foundations of the SC are surprisingly simple: the universe is homogeneous in sufficiently large scales, and it has evolved in a nearly homogeneous way since the cosmic microwave background 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 in inflationary theories, introduced in the early ’80s [5, 16, 17, 18] as a mechanism to solve some of the issues of the SC.
Currently, the inflationary paradigm is one of the most accepted to explain the observed anisotropies in the CMB [19], which were first observed by Penzias and Wilson [20].
There are hundreds of inflation models, in this work we will study a model that stands out for its simplicity and the great agreement with observations, it is the Starobinsky model R + R2 [5], which, as will be seen in the next chapter, can be mapped in one of the best known and also the simplest model, inflation driven by a scalar field.
In this chapter, we summarize the fundamental characteristics of the standard cosmological model in the first section, where also the main issues of SC are described along 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-roll conditions. Finally, the principal results for the cosmological perturbation when inflation 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-ple, which asserts that on sufficiently large scales the universe becomes homogeneous and isotropic [22].
In this case, the spacetime metric can be written as [3,23] ds2 = gµνdxµdxν =−dt2+ a2(t) dr2 1− kr2 + r 2(dθ2+ sin2θdφ2) , (1.1)
where a(t) is the scale factor of the universe, and the constant k is the spatial curvature, which we will take equal to zero throughout this work.
In order to describe the dynamical evolution of the universe, we need to solve Einstein equations, Gµν ≡ Rµν − 1 2δ µ νR = 8πGTµν, (1.2) where Rµ
ν, R are the Ricci tensor and Ricci scalar, respectively, Tµν is the mo-mentum energy tensor and G is the gravitational constant, related with the Planck mass MP through G = MP−1/2, using natural unities. In Eq. (1.2) we have not included a cosmological constant (Λ) term, due to its negligible influence during the inflationary epoch [24].
Using the metric (1.1) in Eq. (1.2), the behavior of the background can be described 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/a is 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 equation of 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 ≡ 3H2MP2 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 universe during the eras dominated by radiation and matter, that is to say, from 10−2 s after the big-bang until now. Some of the most important predictions of this model are [1, 25]: the abundance of light elements, the existence of a cosmic background radiation with a Planckian spectrum at some temperature different from zero, and the explanation of gravitational lens systems.
Despite the success of the SC, there are some questions and problems that can not be explained within the model, and some of them will be summarized below.
The fundamental principles of homogeneity and isotropy can not be explained inside this framework without violating the causality principle,it is what defines the so 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 the scale factor obtained from the solution of Eqs. (1.3), it can be shown that Ω = 1 is a point of unstable equilibrium of Eq. (1.6) for ω ≥ 0 (which is the case for radiation and matter dominated eras), due to the fact that the term on the right hand side of Eq. (1.6) always increases. This is known as the flatness problem.
Another problem with the SC concerns the abundance of topological defects, such as magnetic monopoles, cosmic strings, or string-wall systems that may arise from the spontaneous breaking of symmetries assumed in the different particle physics models interacting in the FRW space-time, and which are not currently observed [26].
Finally, the question about the origin of primordial fluctuations [3], the seeds for the large structure, galaxies, clusters, and so on, that can be seen today, has no natural explanation in this theory.
The causes of the problems previously mentioned are related to the fact that both 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 universe would 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 universe originates from a causally connected region. On the other hand, it is clear that the flatness problem could be solved, because the term to the right of (1.6), would be decreasing at first, and then will be increasing in FRW cosmology dominated by radiation or matter, until reaching its current value.
Finally, any density of unwanted relics would be diluted by inflation and the anisotropies of the CMB, the origin of the large structure would be explained by fluctuations of the field driving inflation.
1.2
Inflationary model
In simple terms, we can describe an inflationary era as any period in which the scale factor of the Universe, a(t), is growing at an accelerated rate ¨a(t) > 0, which can be written 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 satisfied during the inflationary period. As ρ > 0, then we need a negative pressure in order to accomplish the inflationary conditions [27].
In order to describe an inflationary epoch in the early universe, a plethora of inflationary models has been proposed. There are models with scalar and vector fields to drive inflation [28, 29], scalar-multi-tensorial theories [30], or inflationary models appearing as the consequence of modifications to the Einstein Hilbert la-grangian (f(R) theories), wich lead to higher derivative terms in the equation of motion [5, 31, 32], among other possibilities. An important point to be remarked here is that several of these models are dynamically equivalent to theories with scalar fields driving inflation.
In particular, higher derivative theories [9, 33], the object of study of the first part 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 the following.
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 a sufficiently flat region to develop the slow roll regime [35], to be defined later.
In a toy model with effective potential V (φ) = m2
2 φ
2and mass m, several regimes are possible depending on the value of the field φ: for sufficiently small values of V (φ) small field fluctuations are generated, and this field moves slowly towards the minimum of its potential; while for values close to the minimum of V (φ), the scalar field undergoes rapid oscillations, creating pairs of particles and warming up the universe. Since this potential function has a minimum at φ = 0, it could be expected that the scalar field φ oscillates around this minimum. If the Universe does not expand, the equation of motion for φ in this regime is similar to the harmonic oscillator case, ¨φ = −m2φ [36].
In the general case, a new term 3H ˙φ appears due to the expansion of the universe, modifying the equation of motion obtained from the Robertson-Walker metric [37] as
¨
φ + 3H ˙φ + V0(φ) = 0, (1.9)
which, for the special case of a quadratic potential considered before, becomes: ¨
φ + 3H ˙φ =−m2φ, (1.10)
and the term 3H ˙φ 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 from the lagrangian (1.8) as [40]
ρ = 12φ˙2+ V (φ), (1.11)
p = 12φ˙2 − V (φ), (1.12)
1That is to say, we will not include terms proportional to Rφ in the Lagrangian density, where
and from these expressions we can see that, if the potential energy is much larger than the kinetic energy, we have that p ≈ −ρ, in such a way that the inflationary condition is satisfied.
From Einstein equations it can be shown that the energy density ρ, and therefore the Hubble parameter H, are constant in this case [38], leading to the exponential dependence 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 that the field is rolling slowly towards the minimum of its potential [43]
¨
φ 3H ˙φ, φ˙2 V (φ), (1.14)
in such a way that the Einstein equation and the equation of motion for the φ field can be written as
H2 ≈ 3M8π2 P
V (φ), 3H ˙φ ≈ −V0(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:
= M 2 P 2 V0 V 2 , η = MP2V 00 V . (1.16)
Using (1.15) and (1.16) we can note that the slow roll conditions (1.14) are equivalent to [48]:
1, |η| 1. (1.17)
These parameters will be used to determine observables such as the spectral index ns 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]. Potentials
satisfyinghV − φV0i > 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 the
conditions 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.
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 in terms of the Hubble parameter or the scalar potential as [49]
N (t) = Z tf t Hdt = Z φf φ H ˙ φdφ≈ − 1 M2 p Z φf φ V V0dφ. (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 conditions are violated, ≈ 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), there are quantum fluctuations. Due to the fact that the energy density is dominated by 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 FRW metric.
The primordial scalar (PR(k))and tensor (PT(k))spectra can be written in terms of the Hubble parameter as
PR(k) = Hk2 2π ˙φc 2 , PT(k) = 16H 2 k πMP2 , (1.21)
where the Hubble parameter Hk and the φc field are evaluated at the time when the perturbation of momentum k leaves the horizon [52].
From Eqs. (1.14) and (1.15), it is clear that the primordial spectra depend on the 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 slowly with time during the inflation era, the spectra are nearly scale-invariant, that is to say, 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 and tensor spectral indices, respectively, which can be written in terms of the slow roll parameters (1.16) [53] as
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 as r = 16.
In the next chapter we are going to study one of the most appealing inflationary model, the R + R2 model, a particular case of the f(R) theories, first proposed by Starobinsky [5], and we will show that when the RR term is included as a small perturbation to R + R2, the spectral index n
s and the ratio tensor to scalar spectra r are consistent with experimental constraints.
Chapter 2
On higher derivative corrections to
the R
+ R
2
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 scalar potential on which this field evolves in time.
In this chapter, we will present the results published in [12]. In that paper, we analyzed the characteristics of the R + R2 model of inflation when the term RR is included as a small perturbation.
In the previous chapter, we remarked that this kind of model leads to higher derivative terms in the equations of motion. The role of higher derivative terms can be seen from different perspectives. In the semiclassical approach to gravity, these terms are required to lead to a renormalizable theory [54] (see [55] for a review and further references).
The same situation holds in quantum gravity, where fourth derivative terms provide renormalizability [56]. An important advantage of higher derivative terms is that their effects are strongly suppressed below the Planck scale, and hence the classical solutions of general relativity can be seen as a very good approximation at the cosmological, astrophysical and laboratory scales. For this reason, theories with high derivative terms do not contradict the full bunch of experimental and observational tests of Einstein’s gravity.
At low energies, the terms with higher derivatives can be regarded as small perturbations of the fiducial theory of general relativity. This approach has been suggested as an ad hoc universal solution of the ghost problem [57]. This proposal leads to the following dilemma: trying to consider all higher derivative terms as objects to be avoided at the fundamental level and treated as perturbations, one has to ‘forbid’ the R + R2 (Starobinsky) model of inflation [58]. It is easy to note that this is something difficult to accomplish. First of all, the R2-term does not produce ghosts and hence there is no reason to avoid it. At the same time, this inflationary model is the most successful from the observational and phenomenological points of view, so it is not easy to give it up without a real motivation. Finally, the inflation scenario requires the value of the numerical coefficient of the R2-term to be quite big, about 5×108 [10] (see also recent work [59]). This makes the Planck suppression
of this term much less efficient at the scale of inflation. As a result, it would be quite natural to replace the R2-term into another side of the perturbation scheme and 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 the coefficient of the new term.
Previous attempts to analyze higher derivative corrections to the Einstein-Hilbert action have been performed elsewhere [4, 33, 60], making a transformation of the gravitational terms to scalar fields, or with different simplifying assumptions [61,62], which transform the higher derivative theory into a R + R2 theory with modified parameters. Our approach for the analysis of this higher derivative corrections, will be to include them in the lagrangian as small perturbations, and determine their influence 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 section 2.1. In section 2.2, the RR term is included in the standard way, as a new degree of freedom, and, finally, we present the results of treating the term RR as a small perturbation to the Starobinsky model in section 2.3. Conclusions for this chapter are found on section 2.4
2.1
The R
+ R
2Model
Among 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. The action of this model is closely related to vacuum quantum corrections [65, 66] (see also [67,68] and [69] for recent advances in this direction) and, on the other hand, its predictions are consistent with recent bounds including the ones set by the Planck collaboration [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 = MP2
2 Z
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 the scalar degree of freedom (dubbed scalaron in [5]) appearing in f(R) gravity, and we put ~ = c = 1. The theory (2.1) can be easily mapped into a metric-scalar model (see, e.g., [72] and [73] where the procedure is described for the general f(R) extension) S0∗ = M 2 P 2 Z d4x√−gφ0R− U0(φ0) , (2.2)
where the scalar field φ0 is related to the Ricci scalar by the relation
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 field χ0, ¯ gµν = gµν exp nq 2 3 χ0 MP o . (2.5)
The action which results from this procedure has a standard kinetic term, and reads S0∗ = Z d4x√−¯ghM 2 P 2 ¯ R− 1 2( ¯∇χ0) 2− V (χ 0) i , (2.6)
where ( ¯∇χ0)2 = ¯gµν( ¯∇µχ0)( ¯∇νχ0) and V (χ0) is a potential given by the expression V (χ0) = MP2 8α 1− e−√26MPχ0 2 , (2.7)
which drives the evolution of the scalar field χ0 (scalaron) and satisfies the slow roll conditions in the large field regime.
We are interested in considering the modification of the scheme described above when introducing an extra term RR, treated as a perturbation. Before that, in the next section we shall start from a brief review of the standard treatment of the model under discussion, which implies the use of two scalar fields.
2.2
Treating R
R term as a new degree of freedom
In this section we are going to summarize some of the main results obtained when the higher derivative term RR is taken as an extra degree of freedom. In such a case, the standard mapping is done to a theory of two scalar fields.
The new action is
S = M 2 P 2 Z d4x√−gR + 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], and therefore 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 end we write the action as [33] (see also [72])
S = M 2 P 2 Z d4x√−g [F (φ1, φ2) + F1(R− φ1) + F2(R− φ2)] , (2.9)
where F (φ1, φ2) = φ1+ αφ21 + γφ1φ2, F1 = ∂φ∂F1, and F2 = ∂φ∂F2, in such a way that the action takes the form
S = M 2 P 2 Z d4x√−gφ1+ αφ21+ γφ1φ2 + (1 + 2αφ1+ γφ2)(R− φ1) + γφ1(R− φ2)] = M 2 P 2 Z d4x√−g(1 + 2αφ1+ γφ2)R− αφ21 + γφ1R− γφ1φ2] . (2.10)
In order to eliminate the term with R, we integrate by parts [8], Z d4x√−gφ1R = − Z d4x√−g∇µφ1∇µR = Z d4x√−gRφ1, (2.11)
and arrive to the following expression for the action
S = M 2 P 2 Z d4x√−g [(1 + 2αφ1+ γφ2+ γφ1)R − αφ2 1− γφ1φ2 . (2.12)
Defining σ = 1 + 2αφ1 + γφ2 + γφ1, the action can be written in terms of two scalar fields in the form
S = M 2 P 2 Z 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 to perform the conformal transformation of the metric, ¯gµν = e2ϕgµν [75,76],
S = M 2 P 2 Z d4x√−¯ge−4ϕσe2ϕR¯− 6( ¯∇ϕ)2 − 6 ¯ϕ) + γφ1e2ϕ φ¯ 1− 2 ¯∇µϕ ¯∇µφ1 − U(φ1, σ) . (2.14)
In this case, taking σ = e2ϕ, it is straightforward to obtain the action in the Einstein frame S = M 2 P 2 Z d4x√−¯gR¯− 6( ¯∇ϕ)2− γe−2ϕ( ¯∇φ1)2 − U(φ1, ϕ) , (2.15)
where the potential is defined as
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− e 2ϕ), (2.17)
which are consistent with the relation between the original fields φ1 = φ2 and reproduces the results of [9].
From Eq. (2.9), it can be seen that in this case the fields ϕ and φ1 are related with 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 the results 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 the ones presented in [62].
2.2.1
Weak field approximation
The Einstein tensor can be found from equation (2.15), taking the variation with respect to gµν, ¯ Gµν = γe−2ϕ ¯ ∇µφ1∇¯νφ1− 1 2¯gµν ¯ ∇λφ1∇¯λφ1 + 6∇¯µϕ ¯∇νϕ− 1 2g¯µν ¯ ∇λϕ ¯∇λϕ + 1 2¯gµνe −4ϕαφ2 1+ (1− e2ϕ)φ1 . (2.19)
The last expression is consistent with the Einstein tensor obtained for the first time in [9], where the weak field approximation was worked out, showing that the action in this case is given by
Swf ≈ MP2 2 Z d4x√−¯ghR¯− 6 ¯∇λϕ ¯∇λϕ − γ ¯∇λφ1∇¯λφ1− 2φ1ϕ + αφ21 i . (2.20)
The mixed term involving φ1and ϕ can be removed by performing a rotation of these fields, 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 Minkowski space it is necessary that γ < 0, within this approximation. Let us note that this condition does not apply within our approach to the problem.
2.2.2
Simplifying Ansatz
R = r
1R + r
2In 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 general form factors depending on the D’Alambertian operator applied to the Riemann and Ricci tensors and the Ricci scalar.
Our main goal is to show that the application of this simplifying Ansatz to the action (2.15), under certain conditions for the parameters α and γ, reproduces the results 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 = M 2 P 2 Z d4x√−¯gR¯− 6( ¯∇ϕ)2+ γφ1e−2ϕφ¯ 1 − 2γφ1e−2ϕ∇¯µϕ ¯∇µφ1 − U(φ1, ϕ) , (2.22)
or equivalently, using Eq. (2.21), as
S = M 2 P 2 Z d4x√−¯gR¯− 6( ¯∇ϕ)2+ γr1φ21e−4ϕ − U(φ1, ϕ)] . (2.23)
Finally, using the relation between the fields σ = e2ϕ and φ
1, we arrive at the one-scalar representation S = M 2 P 2 Z d4x√−¯gR¯− 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 evolves within 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 in R2 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
ratio r, because the Weyl term C2 is excluded and the non-local operator F
C() is absent [62] in our analysis.
It is important to remark that, as was shown in Refs. [61,62], any solution of the R + R2 theory satisfying the ansatz (2.21), is also a solution of the R + R2+ RR theory, but this is not the whole solution to the theory, as there could be solutions to R+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 term as a small perturbation (as will be done in the following section), in the same spirit in which Ref. [57] analyzes higher order terms as corrections to the Einstein-Hilbert action, leading to the determination of new solutions of the corrected theory. In our case, these solutions are independent of the simplifying ansatz (2.21), but give the same results when this assumption is taken as a particular case.
2.3
Treating R
R term as a small perturbation to
the R
+ R
2model
In the last section we introduced the standard mapping of the R+R2+ RR theory on an inflationary model with two scalar fields, and we learned that this model can be stable in the weak field approximation as long as γ < 0, and that it is consistent with the simplifying ansatz R = r1R + r2.
Nevertheless, this kind of analysis is not completely free of problems, because the equations of motion following from the action (2.15) do not satisfy the slow roll conditions generically, as discussed in [81]. In this situation, definition of new slow 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 one which 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 standard scheme of mapping to the metric-scalar models (see, e.g., [72]). Instead, we have to follow the treatment of the new term as a perturbation, meaning that the number of 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 mapping to 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 the inclusion of the γ term,
S = S0∗+ Sγ, (2.26) where Sγ is defined as Sγ = M 2 Pγ 2 Z d4x√−gRR R=(φ0−1)2α = M 2 Pγ 8α2 Z d4x√−gφ0φ0. (2.27)
Let us note that the relation between R and φ0 in this formula is exactly the same as the obtained for the unperturbed R + R2 model given by equation (2.3) without any changes. This procedure means that we disregard all possible terms of higher orders in γ.
In order to obtain the scalar mapping under this approximation, we use the relation between the σ = e2ϕ and φ
1 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(e
2ϕ
− 1) − 2αγ2e2ϕ. (2.28)
When Eq. (2.28) is substituted back into the action, this gives us
S = M 2 P 2 Z d4x√−¯g h ¯ R− 6( ¯∇ϕ)2 1 + ke2ϕ− U(ϕ) i , where k = γ 6α2 and U(ϕ) = e−4ϕ 4α (e 2ϕ − 1)2 (2.29)
which has exactly the same form as the one in Eq. (2.25). Nevertheless, in this case the kinetic term has a non-canonical form. Thus, to find the mass of the field in the Minkowski limit, it has to be transformed into the standard form.
It is important to emphasize that all the analyses made in this work are concerned with the case |k| 1, in order to allow the treatment of the R term as a small perturbation to the R + R2 theory.
It is easy to see that when the Ansatz (2.21) is used alongside with the previous approximation, we recover the results of [61, 62]. In order to check this, we write the action (2.29) in the equivalent way,
S = M 2 P 2 Z d4x√−¯ghR¯− 6( ¯∇ϕ)2 + 3k(σ− 1) ¯ϕ − U(ϕ)i, where σ = e2ϕ, (2.30)
as defined above. Now, we can use Eq. (2.28) to find the relation 1− γ α −1h r1+ σ− γ α(σ) i = r1σ. (2.31)
When expanded up to the linear order in the operator γ
α we arrive at the relation σ = r1(σ− 1).
With these considerations, the action (2.30) becomes S = M 2 P 2 Z d4x√−¯gR¯− 6( ¯∇ϕ)2− U(ϕ), (2.32) where the potential is defined as
U(ϕ) = α− r1γ 4α2 e
Comparing the last expression with the potential corresponding to the nonperturbed Starobinsky model in the Einstein frame given by Eq. (2.7), one can see that the values 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 of the specific Ansatz (2.21) .
Let us stress that although the term γRR studied here is a particular case of 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 this work is not a particular case of that in [61, 62], but represents a qualitatively new approach to the RR term in the action.
The bilinear part of the action (2.30) can be cast into the form
S = Z d4x√−¯gnM 2 P 2 ¯ R− 1 2( ¯∇χ) 2 − 1 2M 2 χχ2 o , (2.34)
where fields χ and ϕ are related by the relation
ϕ = p χ
6 (1 + γ/6α2) M P
. (2.35)
Let k ≡ γ
6α2. If |k| 1, the mass of the field χ is Mχ2 ≈ 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 smaller than M depending on the sign of the parameter γ. This behavior is preserved by small values of |k| 1 in the general case of (2.29), as it will be seen from the estimated values for this quantity obtained using the Planck collaboration results [13].
Returning to the general expression for the action (2.29), the field transformation that turns the kinetic term in a canonical form should satisfy
dχ dϕ
2
= 6MP2 1 + ke2ϕ, (2.37)
and then the action becomes S = Z d4x√−¯gnM 2 P 2 ¯ R− 1 2( ¯∇χ) 2− V (ϕ(χ))o. (2.38)
In the Einstein frame, the potential is given by V (χ) = V (ϕ(χ)) = M
2 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 transcendental equation that follows from (2.37),
χ √ 6MP = ln " 1−√1 + ke2ϕ eϕ(1−√1 + k) # +√1 + ke2ϕ−√1 + k. (2.40)
0 0.5 1 1.5 2 2.5 3 3.5 −0.050 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 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 √ 6 M P k e 2 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 √ 6 M P k e 2 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 perturbative parameter k. The shaded regions are excluded by the constraints imposed by Planck data [13], as will be shown later.
In the last expression for the particular case ke2ϕ ≈ 1, and taking into account that |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 be seen from figure 2.1, where we have plotted Eq. (2.41) which shows the behavior of the new scalaron χ in the particular case ke2ϕ ≈ 1, where the shaded regions are excluded by the Planck data, as will be discussed below. Let us note that the condition ke2ϕ ≈ 1 has been chosen since it corresponds to the maximal value of the parameter k which is compatible with our approximation |k| 1. For smaller values 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 perturbative approach 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 the parameter k, with k = 0 corresponding to the R+R2 model, and the other values to the extra RR term with different coefficients. One can observe that the presence of the 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 than for the standard R + R2 model.
For positive k, all expanding spatially-flat FRW universes evolve to the dust-like one filled by massive scalarons at rest at late times, like in the case of the Starobinsky model. As we will see later, at earlier times, they can develop inflation in the slow roll regime. For the case of negative values of k (remember |k| 1), there is a
maximum in the potential for a critical value of the χ field, given by the expression χmax √ 6MP = ln p |k| 1−p1− |k| ! −p1− |k|, (2.42)
which comes from the constraint of a real χ field in Eq. (2.40). In this case, slow roll inflation can take place only for values of k which are close to zero, as will be seen from the behavior of the slow roll parameters and η. In fact, this is not too relevant, 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 the last 60 e-folds of inflation begin, leading to a modification of observable parameters such 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 its minimum, and that can affect the oscillations of the field χ (new scalaron) after inflation and gravitational creation of particles and antiparticles by these oscilla-tions through parametric resonance [5, 85, 86, 87]. The next important question is whether a non-zero γ modifies the conditions of a slow roll inflation.
8α M2 PV (χ) χ √ 6MP k =−0.1 k =−0.07 k = 0.0 k = 0.25 k = 0.5
Figure 2.2: RR correction to the Starobinsky model of inflation for different values of 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 derivative of the Hubble parameter H has to be sufficiently small. As a result, the slow roll parameters =− ˙ H H2, η = − ˙ 2H =− ¨ χ H ˙χ, (2.43)
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 terms of the potential as = M 2 P 2 V0(χ) V (χ) 2 and η = M2 P V00(χ) V (χ) (2.44)
and, using Eq. (2.40), the two parameters can be written in terms of the field ϕ,
= 4 3 1 (1 + ke2ϕ) (1− e2ϕ)2, (2.45) η = −4 3 " e−2ϕ(1− 2e−2ϕ) + k2 (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 where the slow roll regime is valid: when |k| 1. For k < 0, we have a slow roll regime for 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 M2 P Z χ χe V (χ) V0(χ)dχ, (2.47)
where χe corresponds to the end of inflation. In terms of the field ϕ we get N (ϕ) = 3 4 h − 2ϕ + (1 − k) e2ϕ+ k 2e 4ϕi 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 2e 4ϕ . (2.49)
One can use this relation to derive the value ϕN of the field ϕ, corresponding to the instant when the universe expanded by N e-folds,
ϕN ≈ 1 2ln " 1− 1 k ± r 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 + r 1 k2 + 8N 3k # . (2.51)
0 0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 S lo w ro ll p ar am et er ǫ χ √ 6MP l k =−0.1 k =−0.07 k = 0.0 k = 0.25 k = 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 S lo w ro ll p ar am et er η χ √ 6MP k =−0.1 k =−0.07 k = 0.0 k = 0.25 k = 0.5
Figure 2.3: The slow roll parameters (up) and η (down) for the model with RR term as functions of the inflaton field χ.
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 its derivatives at the moment when their physical wavelength λ = p/a(t), p = const crosses the Hubble radius H−1(t) during inflation. The same matching condition helps to express N as a function of the present physical scale λ = a(t0)/p where t0 is the present moment. Then one gets the standard expressions for the spectral index ns(p) of the power spectrum of primordial curvature perturbations and the tensor-to-scalar ratio r(p) in the leading order of the slow-roll approximation:
ns− 1 = −6 + 2η = MP2 " 2V 00 V − 3 V0 V 2# , (2.52) r = 16 =−8MP2 V0 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 to somewhat different physical scales in the Jordan and Einstein frames. Since the standards of length and time intervals are defined in Jordan frame which can be considered as the physical one from the measurement point of view, the number of e-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 the slow-roll approximation (less than a few percent for the model in question), so we may 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(Sq k,N + 3k)2 + 9k [8N (9− 2Sqk,N) + 40Sqk,N− 138] − 54Sqk,N + 18 (Sqk,N − 3)2(Sqk,N+ 3k)2 , r = 576k 2 (Sqk,N− 3)2(Sqk,N + 3k)2 , (2.54)
where we have defined Sqk,N = p
24kN + 9(k− 1)2.
From equations (2.54), we can realize that when k = 0, we recover the predictions of the R + R2 model [10, 88], ns− 1 ≈ − 2 N r≈ 12 N2, (2.55)
and up to the order kN there is no shift in ns− 1 and r, and their corrections are of order k/N or k/N2, respectively.
+ + + > < = = 0.94 0.96 0.98 0. 0.05 0.1 Primordial tilt(ns) Tensor -to -scalar ratio ( r)
Figure 2.4: Observational constraints set by the Planck collaboration [13,71] on the scalar-to-tensor ratio r and scalar spectral index ns, and the prediction for these quantities in the R + R2 + RR model. The allowed regions for this model are presented 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 set by the Planck collaboration [13, 71] is illustrated in Fig. 2.4. The last Planck data constrain 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 the prediction for this quantities in the R + αR2+ γRR model. This figure shows the 68% (dark blue and dark yellow) and 95% (light blue and light yellow) C.L. regions for the measurements of r and ns, taking the combined data as stated in the plot key, and the variation of the R + αR2 model (black line) due to the inclusion of the 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 positive value 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 satisfying the perturbative condition |k| 1 keeps the predictions of the RR model inside the 68% C.L. region. For the case N = 60 the maximum value of k that satisfies this condition is kmax ≈ 0.30. On the other hand, for negative values of k, Planck
