\; CONTENT

THE I.I. MECHNIKOV ODESSA NATIONAL UNIVERSITY

The manuscript

UDC 524.83: 531.51: 530.145

THE COMPACTIFICATION PROBLEMS OF ADDITIONAL DIMENSIONS

IN MULTIDIMENSIONAL COSMOLOGICAL THEORIES

01.04.02 - theoretical physics

The PhD thesis for

physical and mathematical science

Zhuk Alexander Ivanovich

Dr. of phys.-math. sci., Professor

Odessa – 2011

You can’t look forward with head down.

A.A. Cron

## List of Abbreviations

– ADditional Dimensions model (or large extra dimensions model);

– Anti de Sitter;

BBN

– Big Bang Nucleosynthesis;

CDM

– Cold Dark Matter with Lambda-term;

CMB

– Cosmic Microwave Background;

ETG

– Extended Theory of Gravitation;

GR

– General Relativity;

KK

– Kaluza-Klein;

MCM

– Multidimensional Cosmological Model.

## Introduction

Latest observation of Ia type supernovas and CMB yields to following composition of the Universe: 4% of baryonic matter, 20% of dark matter and 76% of dark energy. The term ”dark matter” is implemented for the unknown matter, which has an ability for clustering, but was not yet detected in lab-conditions. The ”dark energy” is the energy not been detected yet, but also unable for clustering, as common energy does.

Most probably, the dark energy is responsible for the value of the cosmological constant. Recent experiments indicate the value of the cosmological constant to be too small if originated only by vacuum energy of common matter. This brings some difficulties for developing of corresponding theories. A few ways for interpretation of presents for the cosmological constant are known. Some of those are dynamical alternatives of dark energy.

In this Thesis the attention is focused on theory. This kind of gravitational theory is the result of the lagrangian generalization in the Hilbert-Einstein action. On the base of this theory a study will be undertaken on the problem of the effective cosmological constant, accelerated expansion of the Universe and the compactification (non-observation) of additional dimensions.

Topicality of the subject.

Multidimensionality of our Universe is one of the most intriguing assumption in modern physics. It follows naturally from theories unifying different fundamental interactions with gravity, e.g. M/string theory [1, 2]. The idea has received a great deal of renewed attention over the last few years. However, it also brings a row of additional questions.

According to observations the internal space should be static or nearly static at least from the time of primordial nucleosynthesis, otherwise the fundamental physical constants would vary. This means that at the present evolutionary stage of the Universe there are two possibilities: slow variation or compactification of internal space scale parameters [3].

in many recent studies the problem of extra dimensions stabilization was studied for so-called ADD (see e.g., Refs. [4, 5, 6, 7, 8, 9, 10, 11]). Under these approaches a massive scalar fields (gravitons or radions) of external space-time can be presented as conformal excitations.

In above mentioned works it was assumed that multidimensional action to be linear with respect to curvature. Although as follows from string theory, the gravity action needs to be extended to nonlinear one. In order to investigate effects of nonlinearity, in this Thesis a multidimensional Lagrangian will be studied, having the form , where is an arbitrary smooth function of the scalar curvature.

Connection of the Thesis with scientific programs,

plans and projects.

The study under this Thesis was undertaken as part of the budget project of Odessa National Mechnikov University #519 ”Compactification of space-time in quantum cosmology”.

The target and the task of the study.

The task of this study is an investigation of diverse non-linear cosmological models with respect to condition of additional dimensions compactification and an analysis of additional dimensions effect on the evolution of the Universe.

For that, several surveys have to be implemented: an analysis of non-linearity influence on the effective potential in the equivalent theory; capability of extrema for these potentials as result of additional dimensions non-observability condition. Stabilization and compactification of additional dimensions and their consistency with observations.

Hence, the attention is accumulated on the possibility to achieve models, in which external space (our 4D space-time) behaves as being observed Universe and internal space is stabilized and compactified on the Plank scales. These kind of models allow to build a bridge between observations, multidimensionality and evolution of the Universe, proving by this the principal possibility for existence of additional dimensions.

The scientific novelty of obtained results.

The multidimensional models of type for pure geometric action and of with forms are studied for stabilization and compactification of additional dimensions.

It is shown, that for non-linearity of type in multidimensional case with forms, the stabilization of internal spaces is independent on signatures of internal space curvature, multidimensional and effective cosmological constants. Moreover, effective cosmological constant may satisfy to observed densities of dark energy by applying fine tuning. First time for this kind of model, the capability for stable compactification of additional dimensions and accelerated expansion of the Universe is shown. When in case of pure geometrical model of this two phenomena simultaneously are not obtainable.

The analysis of inflation is undertaken for linear, quadratic, quartic models with forms and for model with pure gravitational action.

A new type of inflation is found for model, called as bouncing inflation. For this type of inflation there is no need for original potential to have a minimum or to check the slowroll conditions. A necessary condition is the existence of the branching points. It is shown that this inflation takes place both in the Einstein and Brans-Dicke frames.

Practical value of obtained results.

The obtained results have fundamental and theoretical value, giving possibility for assay of the Universe evolution, taking into account presence of additional dimensions for diverse non-linear models either pure gravitational or with forms. The parameters values obtained in the Thesis may become an object for further study and justification in experiments.

Approbation of the Thesis results.

The main results of the study, present in the Thesis, were reported on following conferences:

1. International conference ”QUARQS 2008”, May 23-26, Ser.-Pasad, Russia;

2. The 8-th G. Gamow Odessa International Astronomical Summer School, Astronomy and beyond: Astrophysics, Radio astronomy, Cosmology and Astrobiology, August 1-5, 2006, Odessa, Ukraine;

3. The 6-th G. Gamow Odessa International Astronomical Summer School, Astronomy and beyond: Astrophysics, Radio astronomy, Cosmology and Astrobiology, August 1-5, 2006, Odessa, Ukraine;

4. VI International Conference: Relativistic Astrophysics, Gravitation and Cosmology, May 24-26 2006, Kyiv, Ukraine;

5. Kyiv High Energy Astrophysics Semester (by ISDC, CERN), April 17 - May 12, 2006, Kiyv, Ukraine;

6. III International Symposium Fundamental Problems in Modern Quantum Theories and Experiments, September 2005, Kiyv-Sevastopol, Ukraine.

Publications.

The main results of the study, present in the Thesis, were explained in following publications:

1. Saidov T. 1/R multidimensional gravity with form-fields: Stabilization of extra dimensions, cosmic acceleration, and domain walls / Saidov T., Zhuk. A. // Phys. Rev. D. – 2007. – V. 75. – 084037, 10 p.

2. Saidov T. AdS Nonlinear Curvature-Squared and Curvature-Quartic Multidimensional (D=8) Gravitational Models with Stabilized Extra Dimensions / Saidov T., Zhuk. A. // Gravitation and Cosmology. – 2006. – V. 12. – P. 253–261.

3. Saidov T. A non linear multidimensional gravitational model with form fields and stabilized extra dimensions / Saidov T., Zhuk. A. // Astronomical & Astrophysical Transactions. – 2006. – V. 25. – P. 447–453.

4. Saidov T. Problem of inflation in nonlinear multidimensional cosmological models / Saidov T., Zhuk. A. // Phys. Rev. D. – 2009. – V. 79. – 024025, 18 p.

5. Saidov T. Bouncing inflation in a nonlinear gravitational model / Saidov T., Zhuk. A. // Phys. Rev. D. – 2010. – V. 81. – 124002, 13 p.

## Chapter 1 Literature Overview

The currently observed accelerated expansion of the Universe suggests that cosmic flow dynamics is dominated by some unknown form of dark energy characterized by a large negative pressure[12]. Most recent observations of CMB and Ia supernovas indicates that Universe consists of: 76% of dark energy, 20% of dark matter and 4% of baryonic matter [13, 14, 15, 16]. Many approaches are known for distinguishing of dark matter and dark energy. Usually it is assumed that dark energy despite of dark matter is incapable for clustering and satisfies for Strong Energy Condition[17, 18]. Due to dark energy dominates over matter, it is responsible for late-time acceleration of the Universe (i.e. determines the value of effective cosmological constant), following after early-time acceleration as predicted with inflation paradigm [19, 20, 21]. In between these to stages, the period of decelerated expansion is needed to provide eras of radiation domination for BBN and matter domination for formation of Universe structure

As general approach, the standard Einstein’s General Relativity (GR) is used as paradigm (foundation) for developing of new theories, called as Extended Theories of Gravity (ETG). This allows to keep successful features of GR and apply extensions and corrections to Einstein’s theory. Usually by adding higher order curvature invariants and/or minimally or non-minimally coupled scalar fields to the dynamics; these corrections emerge from the effective action of quantum gravity [12, 22].

Among many, the simplest theory capable more or less to describe experimental data is so-called CDM model. It gives appropriate qualitative picture of the observed Universe, but do not explain the inflation. Also, in different models well known problem of the cosmological constant magnitude arises being higher in 120 orders of magnitude then observed one [23].

As next step for modification of GR, one may apply Mach’s principle, which states that the local inertial frame is determined by the average motion of distant astronomical objects [24]. This means that the gravitational coupling could be determined by the distant distribution of matter, and it can be scale-dependent and related to some scalar field. As a consequence, the concept of ”inertia” and the Equivalence Principle have to be revised. Brans-Dicke theory [25] constituted the first consistent and complete theory alternative to Einstein’s GR. Brans-Dicke theory incorporates a variable gravitational coupling strength whose dynamics are governed by a scalar field non-minimally coupled to the geometry, which implements Mach’s principle in the gravitational theory [12, 25, 26, 27].

Nonlinear ETG models may arise either due to quantum fluctuations of matter fields including gravity [28], or as a result of compactification of extra spatial dimensions [29]. Compared, e.g., to others higher-order gravity theories, theories are free of ghosts and of Ostrogradski instabilities [30]. Recently, it was realized that these models can also explain the late time acceleration of the Universe. This fact resulted in a new wave of papers devoted to this topic (see e.g., recent reviews [31, 32, 12]).

Another intriguing assumption in modern physics is the multidimensionality of our Universe. It follows from theories which unify different fundamental interactions with gravity, such as M or string theory [1, 2], and which have their most consistent formulation in spacetimes with more than four dimensions. Thus, multidimensional cosmological models have received a great deal of attention over the last years.

Stabilization of additional dimensions near their present day values (dilaton/geometrical moduli stabilization) is one of the main problems for any multidimensional theory because a dynamical behavior of the internal spaces results in a variation of the fundamental physical constants. Observations show that internal spaces should be static or nearly static at least from the time of recombination (in some papers arguments are given in favor of the assumption that variations of the fundamental constants are absent from the time of primordial nucleosynthesis [33]). In other words, from this time the compactification scale of the internal space should either be stabilized and trapped at the minimum of some effective potential, or it should be slowly varying (similar to the slowly varying cosmological constant in the quintessence scenario). In both cases, small fluctuations over stabilized or slowly varying compactification scales (conformal scales/geometrical moduli) are possible.

Stabilization of extra dimensions (moduli stabilization) in models with large extra dimensions (ADD-type models) has been considered in a number of papers (see e.g., Refs. [4, 11]). In the corresponding approaches, a product topology of the dimensional bulk spacetime was constructed from Einstein spaces with scale (warp) factors depending only on the coordinates of the external dimensional component. As a consequence, the conformal excitations had the form of massive scalar fields living in the external spacetime. Within the framework of multidimensional cosmological models (MCM) such excitations were investigated in [34, 35] where they were called gravitational excitons. Later, since the ADD compactification approach these geometrical moduli excitations are known as radions [4, 6].

Most of the aforementioned papers are devoted to the stabilization of large extra dimensions in theories with a linear multidimensional gravitational action. String theory suggests that the usual linear Einstein-Hilbert action should be extended with higher order nonlinear curvature terms. In the papers [36, 37] a simplified model is considered with multidimensional Lagrangian of the form , where is an arbitrary smooth function of the scalar curvature. Without connection to stabilization of the extra-dimensions, such models (dimensional as well as multidimensional ones) were considered e.g. in Refs. [38, 39, 40, 41, 42, 43, 44, 45, 46]. There, it was shown that the nonlinear models are equivalent to models with linear gravitational action plus a minimally coupled scalar field with self-interaction potential. Similar approach was elaborated in Refs. [47] where the main attention was paid to a possibility of the late time acceleration of the Universe due to the nonlinearity of the model.

The most simple, and, consequently, the most studied models are polynomials of : , e.g., quadratic and quartic ones. Active investigation of these models, which started in 80-th years of the last century [48, 49, 50], continues up to now [51, 52]. Obviously, the correction terms (to the Einstein action) with give the main contribution in the case of large , e.g., in the early stages of the Universe evolution. As it was shown first in [53] for the quadratic model, such modification of gravity results in early inflation. From the other hand, function may also contain negative degrees of . For example, the simplest model is . In this case the correction term plays the main role for small , e.g., at the late stage of the Universe evolution (see e.g. [37, 54] and numerous references therein). Such modification of gravity may result in the late-time acceleration of our Universe [55]. Nonlinear models with polynomial as well as -type correction terms have also been generalized to the multidimensional case (see e.g., [37, 54, 56, 57, 58, 59, 60, 61]).

In this Thesis several models of -theory are analyzed for pure gravitational case in chapter 2, and with forms in chapter 3, as well as inflation possibility is studied in chapter 4.

## Chapter 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS

In this chapter the non-linear gravitational models with curvatures of type and are studied. It is shown that for particular values of parameters, the stabilization and compactification of additional dimensions is achievable with negative constant curvature of internal space. In this case the dimensional effective cosmological constant becomes negative. As result homogenous and isotropic external time-space appears to be . The correlation between dimensional and dimensional fundamental masses imposes restrictions on the parameters in the models.

2.1. General setup

Let us consider a dimensional nonlinear pure gravitational theory with action functional

 S=12κ2D∫MdDx√|¯g|f(¯R), (2.1)

where is an arbitrary smooth function with mass dimension  ( has the unit of mass) of a scalar curvature constructed from the dimensional metric . is the number of extra dimensions and denotes the dimensional gravitational constant which is connected with the fundamental mass scale and the surface area of a unit sphere in dimensions by the relation [62]

 κ2D=2SD−1/M2+D′∗(4+D′). (2.2)

Before endowing the metric of the pure gravity theory (2.1) with explicit structure, let us recall that this nonlinear theory is equivalent to a theory which is linear in another scalar curvature but which contains an additional self-interacting scalar field. According to standard techniques [38, 39, 40, 41, 42, 43, 44, 45, 46, 50], the corresponding linear theory has the action functional:

 S=12κ2D∫MdDx√|g|[R[g]−gabϕ,aϕ,b−2U(ϕ)], (2.3)

where

 f′(¯R)=dfd¯R:=eAϕ>0,A:=√D−2D−1, (2.4)

and where the self-interaction potential of the scalar field is given by

 U(ϕ) = 12(f′)−D/(D−2)[¯Rf′−f], (2.5) = 12e−Bϕ[¯R(ϕ)eAϕ−f(¯R(ϕ))],B:=D√(D−2)(D−1).

This scalar field carries the nonlinearity degrees of freedom in of the original theory, and for brevity will be called the nonlinearity field. The metrics , of the two theories (2.1) and (2.3) are conformally connected by the relations

 gab=Ω2¯gab=[f′(¯R)]2/(D−2)¯gab. (2.6)

Next, let us assume that the D-dimensional bulk space-time undergoes a spontaneous compactification to a warped product manifold

 M=M0×M1×…×Mn (2.7)

with metric

 ¯g=¯gab(X)dXa⊗dXb=¯g(0)+n∑i=1e2¯βi(x)g(i). (2.8)

The coordinates on the dimensional manifold (usually interpreted as our observable dimensional Universe) are denoted by and the corresponding metric by

 ¯g(0)=¯g(0)μν(x)dxμ⊗dxν. (2.9)

For simplicity, the internal factor manifolds are chosen as dimensional Einstein spaces with metrics so that the relations

 Rmini[g(i)]=λig(i)mini,mi,ni=1,…,di (2.10)

and

 R[g(i)]=λidi≡Ri (2.11)

hold. The specific metric ansatz (2.8) leads to a scalar curvature which depends only on the coordinates of the external space: . Correspondingly, also the nonlinearity field depends on only: .

Passing from the nonlinear theory (2.1) to the equivalent linear theory (2.3) the metric (2.8) undergoes the conformal transformation [see relation (2.6)]

 g=Ω2¯g=(eAϕ)2/(D−2)¯g:=g(0)+n∑i=1e2βi(x)g(i) (2.12)

with

2.2. Stabilization of internal dimensions

The main subject of subsequent considerations will be the stabilization of the internal space components. A strong argument in favor of stabilized or almost stabilized internal space scale factors , at the present evolution stage of the Universe, is given by the intimate relation between variations of these scale factors and those of the fine-structure constant [63]. The strong restrictions on variations in the currently observable part of the Universe [64, 65, 66, 67] imply a correspondingly strong restriction on these scale factor variations [63]. For this reason, the derivation of criteria ensuring a freezing stabilization of the scale factors will be performed below.

In Ref. [35] it was shown that for models with a warped product structure (2.8) of the bulk spacetime and a minimally coupled scalar field living on this spacetime, the stabilization of the internal space components requires a simultaneous freezing of the scalar field. Here a similar situation with simultaneous freezing stabilization of the scale factors and the nonlinearity field is expected. According to (2.13), this will also imply a stabilization of the scale factors of the original nonlinear model.

In general, the model will allow for several stable scale factor configurations (minima in the landscape over the space of volume moduli). Let us choose one of them1, denote the corresponding scale factors as , and work further on with the deviations

 ^βi(x)=βi(x)−βi0 (2.14)

as the dynamical fields. After dimensional reduction of the action functional (2.3) one passes from the intermediate Brans-Dicke frame to the Einstein frame via a conformal transformation

 g(0)μν=^Ω2^g(0)μν=(n∏i=1edi^βi)−2/(D0−2)^g(0)μν (2.15)

with respect to the scale factor deviations [36, 37]. As result the following action is achieved

 S=12κ2D0∫M0dD0x√|^g(0)|{^R[^g(0)] − ¯Gij^g(0)μν∂μ^βi∂ν^βj (2.16) − ^g(0)μν∂μϕ∂νϕ−2Ueff},

which contains the scale factor offsets through the total internal space volume

 VD′≡VI×v0≡n∏i=1∫Middiy√|g(i)|×n∏i=1ediβi0 (2.17)

in the definition of the effective gravitational constant of the dimensionally reduced theory

 κ2(D0=4)=κ2D/VD′=8π/M24⟹M24=4πSD−1VD′M2+D′∗(4+D′). (2.18)

Obviously, at the present evolution stage of the Universe, the internal space components should have a total volume which would yield a four-dimensional mass scale of order of the Planck mass . The tensor components of the midisuperspace metric (target space metric on ) reads: , where , see [68, 69]. The effective potential has the explicit form

 (2.19)

where abbreviated

 ^Ri:=Riexp(−2βi0). (2.20)

A freezing stabilization of the internal spaces will be achieved if the effective potential has at least one minimum with respect to the fields . Assuming, without loss of generality, that one of the minima is located at , the extremum condition reads:

 ∂Ueff∂^βi∣∣ ∣∣^β=0=0⟹^Ri=diD0−2(−n∑j=1^Rj+2U(ϕ)). (2.21)

From its structure (a constant on the l.h.s. and a dynamical function of on the r.h.s) it follows that a stabilization of the internal space scale factors can only occur when the nonlinearity field is stabilized as well. In the freezing scenario this will require a minimum with respect to :

 ∂U(ϕ)∂ϕ∣∣∣ϕ0=0⟺∂Ueff∂ϕ∣∣∣ϕ0=0. (2.22)

Hence, a stabilization problem is arrived, some of whose general aspects have been analyzed already in Refs. [34, 35, 36, 37]. For brevity let us summarize the corresponding essentials as they will be needed for more detailed discussions in the next consideration.

1. Eq. (2.21) implies that the scalar curvatures and with them the compactification scales [see relation (2.20)] of the internal space components are finely tuned

 ^Ridi=^Rjdj,i,j=1,…,n. (2.23)
2. The masses of the normal mode excitations of the internal space scale factors (gravitational excitons/radions) and of the nonlinearity field near the minimum position are given as [35]:

 m21 = …=m2n=−4D−2U(ϕ0)=−2^Ridi>0, m2ϕ := d2U(ϕ)dϕ2∣∣∣ϕ0>0. (2.25)
3. The value of the effective potential at the minimum plays the role of an effective 4D cosmological constant of the external (our) spacetime :

 Λeff:=Ueff∫∣∣∣^βi=0,ϕ=ϕ0=D0−2D−2U(ϕ0)=D0−22^Ridi. (2.26)
4. Relation (2.26) implies

 \rm signΛeff=\rm signU(ϕ0)=\rm signRi. (2.27)

Together with condition (2) this shows that in a pure geometrical model stable configurations can only exist for internal spaces with negative curvature2:

 Ri<0(i=1,…,n). (2.28)

Additionally, the effective cosmological constant as well as the minimum of the potential should be negative too:

 Λeff<0,U(ϕ0)<0. (2.29)

Plugging the potential from Eq. (2.5) into the minimum conditions (2.22), (2.25) yields with the help of the conditions

 dUdϕ∣∣∣ϕ0 = A2(D−2)(f′)−D/(D−2)h∣∣∣ϕ0=0, h := Df−2¯Rf′,⟹ h(ϕ0)=0, d2Udϕ2∣∣∣ϕ0 = 12Ae(A−B)ϕ0[∂ϕ¯R+(A−B)¯R]ϕ0 (2.31) = 12(D−1)(f′)−2/(D−2)1f′′∂¯Rh∣∣∣ϕ0>0,

where the last inequality can be reshaped into the suitable form

 f′′∂¯Rh∣∣ϕ0=f′′[(D−2)f′−2¯Rf′′]ϕ0>0. (2.32)

Furthermore, from Eq. (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS) follows

 U(ϕ0)=D−22D(f′)−2D−2¯R(ϕ0) (2.33)

 ¯R(ϕ0)<0 (2.34)

at the extremum.

Thus, to avoid the effective four-dimensional fundamental constant variation, it is necessary to provide the mechanism of the internal spaces stabilization. In these models, the scale factors of the internal spaces play the role of additional scalar fields (geometrical moduli/gravexcitons [34]). To achieve their stabilization, an effective potential should have minima with respect to all scalar fields (gravexcitons and scalaron). Previous analysis (see e.g. [35]) shows that for a model of the form (2.3) the stabilization is possible only for the case of negative minimum of the potential . According to the definition (2.4), the positive branch of will be considered. Although the negative branch can be considered as well (see e.g. Refs. [50, 37, 58]). However, negative values of result in negative effective gravitational ”constant” . Thus should be positive for the graviton to carry positive kinetic energy (see e.g., [12]). From action (2.3) the equation of motion of scalaron field can be obtained in the form:

 to0.0pt$⊓$⊔ϕ−∂U∂ϕ=0. (2.35)

Now, let us analyze the internal space stabilization conditions (2.23) - (2.29) and (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS) - (2.34) on their compatibility with particular scalar curvature nonlinearity .

2.3. model

Recently it has been shown in Refs. [70, 71, 72, 73, 74, 75, 76, 77, 78] that cosmological models with a nonlinear scalar curvature term of the type can provide a possible explanation of the observed late-time acceleration of our Universe within a pure gravity setup. The equivalent linearized model contains an effective potential with a positive branch which can simulate a transient inflation-like behavior in the sense of an effective dark energy. The corresponding considerations have been performed mainly in four dimensions3. Here these analyses are extended to higher dimensional models, assuming that the scalar curvature nonlinearity is of the same form in all dimensions. Starting from a nonlinear coupling of the type:

 f(¯R)=¯R−μ/¯R,μ>0, (2.36)

in front of the term, the minus sign is chosen, because otherwise the potential will have no extremum.

With the help of definition (2.4), the scalar curvature can be expressed in terms of the nonlinearity field and obtain two real-valued solution branches

 ¯R±=±√μ(eAϕ−1)−1/2,⟹ϕ>0 (2.37)

of the quadratic equation . The corresponding potentials

 U±(ϕ)=±√μe−Bϕ√eAϕ−1 (2.38)

have extrema for curvatures [see Eq. (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS)]

 ¯R0,± = ±√μ√D+2D−2 eAϕ0 = 2B2B−A=2DD+2>1forD≥3 (2.39)

and take for these curvatures the values

 U±(ϕ0)=±√μ√D−2D+2e−Bϕ0=±√μ√D−2D+2(2DD+2)−D/(D−2). (2.40)

The stability defining second derivatives [Eq. (2.31)] at the extrema (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS),

 ∂2ϕU±∣∣ϕ0 = ∓√μDD−1√D+2D−2eBϕ0 (2.41) = ∓√μDD−1√D+2D−2(2DD+2)−D/(D−2),

show that only the negative curvature branch yields a minimum with stable internal space components. The positive branch has a maximum with . According to (2.27) it can provide an effective dark energy contribution with , but due to its tachyonic behavior with it cannot give stably frozen internal dimensions. This means that the simplest extension of the four-dimensional purely geometrical setup of Refs. [70, 71, 72, 73, 74, 75, 76, 77, 78] to higher dimensions is incompatible with a freezing stabilization of the extra dimensions. A possible circumvention of this behavior could consist in the existence of different nonlinearity types in different factor spaces so that their dynamics can decouple one from the other. This could allow for a freezing of the scale factors of the internal spaces even in the case of a late-time acceleration with . Another circumvention could consist in a mechanism which prevents the dynamics of the internal spaces from causing strong variations of the fine-structure constant . The question of whether one of these schemes could work within a physically realistic setup remains to be clarified.

Finally, in the minimum of the effective potential , which is provided by the negative curvature branch , one finds excitation masses for the gravexcitons/radions and the nonlinearity field (see Eqs. (2), (2.40) and (2.41)) of order

 m1=…=mn∼mϕ∼μ1/4. (2.42)

For the four-dimensional effective cosmological constant defined in (2.26) one obtains in accordance with Eq. (2.40)  .

Summary

As stability condition the existence of a minimum of the effective potential of the dimensionally reduced theory is assumed, so that a late-time attractor of the system could be expected with freezing stabilization of the extra-dimensional scale factors and the nonlinearity field. It was shown in Refs. [57, 58], that for purely geometrical setups this is only possible for negative scalar curvatures, , independently of the concrete form of the function .

Four-dimensional purely gravitational models with curvature contributions have been proposed recently as possible explanation of the observed late-time acceleration (dark energy) of the Universe [70, 71, 72, 73, 74, 75, 76, 77, 78]. It is shown that higher dimensional models with the same scalar curvature nonlinearity reproduce (after dimensional reduction) the two solution branches of the four-dimensional models. But due to their oversimplified structure these models cannot simultaneously provide a late-time acceleration of the external four-dimensional spacetime and a stabilization of the internal space. A late-time acceleration is only possible for one of the solution branches — for that which yields a positive maximum of the potential of the nonlinearity field. A stabilization of the internal spaces requires a negative minimum of as it can be induced by the other solution branch.

2.4. model

In this section let us analyze a model with curvature-quadratic and curvature-quartic correction terms of the type

 f(¯R)=¯R+α¯R2+γ¯R4−2ΛD. (2.43)

According to eq. (2.4):

 f′=eAϕ=1+2α¯R+4γ¯R3. (2.44)

The definition (2.4) clearly indicates that the positive branch is chosen. For the given model (2.43), the surfaces as a functions are given in Fig. 2.1. As it easily follows from Eq. (2.44), points where all three values and are positive correspond to the region . Thus, this picture shows that there is one simply connected region and two disconnected regions .

Eq. (2.44) can be rewritten equivalently in the form:

 ¯R3+α2γ¯R−14γX=0, (2.45)
 X≡eAϕ−1, −∞<ϕ<+∞ ⟹ −1

Eq. (2.45) has three solutions , where one or three of them are real-valued. Let

 q:=α6γ,r:=18γX. (2.47)

The sign of the discriminant

 Q:=r2+q3 (2.48)

defines the number of real solutions:

 Q>0 ⟹ I¯R1=0,I¯R2,3≠0, Q=0 ⟹ I¯Ri=0 ∀i,¯R1=¯R2, Q<0 ⟹ I¯Ri=0 ∀i. (2.49)

Physical scalar curvatures correspond to real solutions . It is the most convenient to consider as solution family depending on the two additional parameters : , .

For the single real solution is given as

 ¯R1=[r+Q1/2]1/3+[r−Q1/2]1/3:=z1+z2, (2.50)

where is defined in the form:

 z31,2=pe±θ,p2=r2−Q=−q3, cosh(θ)=r√−q3. (2.51)

Taking into account eq. (2.47), the function reads

 X(θ)=8γ√−q3cosh(θ). (2.52)

The three real solutions for are given as

 ¯R1 = s1+s2, ¯R2 = 12(−1+i√3)s1+12(−1−i√3)s2 = ei2π3s1+e−i2π3s2, ¯R3 = 12(−1−i√3)s1+12(−1+i√3)s2 (2.53) = e−i2π3s1+ei2π3s2,

where one can fix the Riemann sheet of by setting in the definitions of

 s1,2:=[r±i|Q|1/2]1/3. (2.54)

A simple Mathematica calculation gives for Vieta’s relations from (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS)

 ¯R1+¯R2+¯R3 = 0, ¯R1¯R2+¯R1¯R3+¯R2¯R3 = −3s1s2=3q, ¯R1¯R2¯R3 = s31+s32=2r. (2.55)

In order to work with explicitly real-valued let us rewrite from (2.54) as follows

 s1,2 = |b|1/3e±iϑ/3, |b|2 = r2+|Q|=r2−Q=−q3, cos(ϑ) = r|b|=r√−q3. (2.56)
 ¯R1 = s1+s2=2|b|1/3cos(ϑ/3), (2.57) ¯R2 = ei2π3s1+e−i2π3s2=2|b|1/3cos(ϑ/3+2π/3), ¯R3 = e−i2π3s1+ei2π3s2=2|b|1/3cos(ϑ/3−2π/3)

or

 ¯Rk = 2|b|1/3cos(ϑ+2πk3) (2.58) = 2√−qcos(ϑ+2πk3),k=−1,0,1.

In order to understand the qualitative behavior of these three real-valued solutions as part of the global solution picture let us first note that, according to (2.45), one may interpret as single-valued function

 X(¯R)=4γ¯R3+2α¯R (2.59)

and look what is happening when are changed. Obviously, the inverse function has three real-valued branches when is not a monotonic function but instead has a minimum and a maximum, i.e. when

 ∂¯RX:=X′=12γ¯R2+2α=0⟹¯R2=−α6γ (2.60)

has two real solutions and corresponding extrema4

 X(¯R±)=43α¯R±. (2.61)

It should hold in this case, so that one find

 γ>0,α<0: Xmax=X(¯R−), Xmin=X(¯R+) γ<0,α>0: Xmax=X(¯R+), Xmin=X(¯R−).

The transition from the three-real-solution regime to the one-real-solution regime occurs when maximum and minimum coalesce at the inflection point

 ¯R+=¯R−=0⟹α=0, γ≠0. (2.63)

(The non-degenerate case is considered here. Models with are degenerated ones and are characterized by quadratic scalar curvature terms only.) Due to the limit where in leading approximation may be considered:

 4γ¯R3≈X→+∞ (2.64)

so that

 ¯R(X→∞)→\rm sign(γ)×∞. (2.65)

Leaving the restriction for a moment aside, it was found that for product there exist three real solution branches :

 γ>0: −∞≤¯¯¯¯¯R1≤¯R−,−∞≤X≤Xmax, ¯R−≤¯¯¯¯¯R2≤¯R+,Xmax≥X≥Xmin, ¯R+≤¯¯¯¯¯R3≤+∞,Xmin≤X≤+∞, γ<0: −∞≤¯¯¯¯¯R1≤¯R−,+∞≥X≥Xmin, ¯R−≤¯¯¯¯¯R2≤¯R+,Xmin≤X≤Xmax, ¯R+≤¯¯¯¯¯R3≤+∞,Xmax≥X≥−∞.

It remains for each of these branches to check which of the solutions from (2.58) can be fitted into this scheme. Finally, one will have to set the additional restriction on the whole picture.

To define the conditions for minima of the effective potential , first the extremum positions of the potential is to be obtained. The extremum condition (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS) for the given particular model (2.43) reads:

 γ¯R4(0)i(D2−4)+α¯R2(0)i(D2−2)+¯R(0)i(D2−1)−DΛD=0, (2.67)

2.5. Case - analytical solution

In this part let us investigate the case of positive that is equivalent to the condition

 Q(ϕ)>0⇒\rm signα=\rm signγ. (2.68)

The case that corresponds to different signatures of the discriminant will be considered in chapter 4.

To define the conditions for minima of the effective potential , first let us obtain the extremum positions of the potential . The extremum condition (CHAPTER 2 ASYMPTOTICALLY AdS NONLINEAR GRAVITATIONAL MODELS) for the given particular model (2.43) reads:

 ¯R4(0)1γ(D2−4)+¯R2