Compact extra dimensions in cosmologies with f ðTÞ structureFranco Fiorini,1,2,* P. A. González,3,† and Yerko Vásquez4,5,‡

1Centro Atómico Bariloche, Comisión Nacional de Energia Atómica R8402AGP,8400 Bariloche, Argentina

2Sede Andina, Universidad Nacional de Río Negro, Villegas 137, 8400 Bariloche, Argentina3Facultad de Ingeniería, Universidad Diego Portales, Avenida Ejército Libertador 441,

Casilla 298-V, Santiago, Chile4Departamento de Física, Facultad de Ciencias, Universidad de La Serena,

Avenida Cisternas 1200, La Serena, Chile5Departamento de Ciencias Físicas, Facultad de Ingeniería, Ciencias y Administración,

Universidad de La Frontera, Avenida Francisco Salazar 01145, Casilla 54-D, Temuco, Chile(Received 30 July 2013; published 22 January 2014)

The presence of compact extra dimensions in cosmological scenarios in the context of fðTÞ-like gravitiesis discussed. For the case of toroidal compactifications, the analysis is performed in an arbitrary number ofextra dimensions. Spherical topologies for the extra dimensions are then carefully studied in six and sevenspacetime dimensions, where the proper vielbein fields responsible for the parallelization process are found.

DOI: 10.1103/PhysRevD.89.024028 PACS numbers: 04.50.Kd, 04.50.-h, 98.80.-k

I. INTRODUCTION

In 1928 Einstein proposed an equivalent formulation ofgeneral relativity (GR) nowadays known as the telepar-allel equivalent of general relativity (TEGR) [1,2]. In thistheory, the Weitzenböck connection is used instead ofthe Levi-Cività connection to define the covariant deri-vative. Weitzenböck connection has non-null torsion;however, it is curvatureless, which implies that thisformulation of gravity exhibits only torsion. In D space-time dimensions, the dynamical fields are the D linearlyindependent vielbeins and the torsion tensor is formedsolely by them and the first derivatives of these objects.The Lagrangian density, which will be noted hereafter asT, is constructed from this torsion tensor, assuminginvariance under general coordinate transformations andlocal Lorentz transformations, along with demanding thatthe Lagrangian density be quadratic in the torsion tensor[2]. In recent years an extension of the above Lagrangiandensity was constructed in [3–6], making the Lagrangiandensity a function of the scalar torsion T, the so-calledfðTÞ gravity. Much attention has been focused on thisextended fðTÞ gravity theory recently, because it exhibitsinteresting cosmological implications, as witnessed in thestudy of missing matter problems, as well as providing anew mechanism to explain the late acceleration of theUniverse based on a modification of the gravitationaltheory, instead of introducing an exotic content of matter(dark energy problem) [7–15,17,16,18]. It is also believedthat fðTÞ gravity could be a reliable approach to addressthe shortcomings of general relativity at high-energy

scales [19]. For instance, in [4] a Born-Infeld fðTÞgravity Lagrangian was used to cure the physicallyinadmissible divergencies occurring in the big bang ofstandard cosmology, rendering spacetime geodesicallycomplete and powering an inflationary stage withoutthe introduction of an inflaton field.We are now interested in studying this alternative theory

of gravity in a higher-dimensional context, and the basicfeatures of cosmological behavior due to this extension.The conception of extra dimensions in physical theories hasa long history that begins with the original ideas of Kaluzaand Klein [20] and finds a new realization nowadays inmodern string theory [21,22]. One of the main motivationsfor studying these higher-dimensional models is the chanceof unifying the fundamental interactions of nature. In mostextra-dimensional scenarios, the additional dimensions areassumed to be compactified on a very small (unobservable)internal submanifold, whereas the other spacetime dimen-sions constitute the four-dimensional observable universe.Another mechanism of dimensional reduction studied incosmological GR scenarios is dynamical compactification[23–25]. In this case, the internal dimensions evolve in timeto very small scales as the external dimensions expand, andso the observable universe becomes effectively four dimen-sional. This type of reduction was first studied by Chodosand Detweiler in [23], who considered vacuum Einsteinequations in five spacetime dimensions. These results werelater extended, and a full classification of homogeneous 11-dimensional cosmologies can be found in [26]. Dynamicalreduction in multidimensional Bianchi type I models wasstudied in [27] and a study of multidimensional cosmo-logical models as dynamical systems with topologyFRW 4 × TD−4 [here TD−4 refers to the ðD − 4Þ torus]was performed in [28,29]. More recently, and in a totally

*francof@cab.cnea.gov.ar†pablo.gonzalez@udp.cl‡yvasquez@ufro.cl

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different context, GR in the presence of a large number ofextra dimensions was carefully studied in [30,31].The specific topic of extra dimensions has scarcely been

worked out in fðTÞ theories, and it remits, on one hand,to some black hole solutions of fðTÞ gravity in arbitraryspacetime dimensions [32], and on the other hand, to ananalysis of four-dimensional fðTÞ gravities as effectivetheories coming from five-dimensional Kaluza-Klein andRandall-Sundrum models [33]. In this work we investigatefor the first time cosmological solutions in the context offðTÞ gravity in higher dimensions, considering that theextra dimensions are compactified in different ways. Thisinvestigation is motivated primarily by the increasinginterest in multidimensional cosmological scenarios, par-ticularly brane-world models [34]. In the present work weshall consider both toroidal and spherical topologies forthe internal dimensions. With this purpose in mind weproceed first to obtain the proper parallel vector fields forthe different topologies under consideration. This topic is adifficult one, and constitutes the starting point of any modeldescribing extra dimensions in gravitational theories withabsolute parallelism, fðTÞ gravity being one of them. Inparticular we consider five-, six-, and seven-dimensionalmodels and study all the possible compactifications of theextra dimensions using the right vielbein field that paral-lelizes the spacetime in each of these cases, and we thenproceed to obtain the corresponding field equations. Thenwe show in general terms, that in some limited cases,simple analytical solutions describe physically relevantspacetimes representing different stages of cosmic evolu-tion. The results presented here are mandatory aspectsrequired to deal with more fundamental topics of physicalcosmology such as structure formation and cosmologicalperturbations.The paper is organized as follows: In Sec. II we present

a brief review of the fundamentals of fðTÞ theories. InSec. III we study toroidal compactification for an arbitrarynumber of extra dimensions, and in Sec. IV we studyspherical compactifications with the emphasis on six andseven spacetime dimensions. Finally, in Sec. V we presentthe conclusions.

II. FUNDAMENTALS OF f ðTÞ THEORIES

The extended gravitational schemes with absoluteparallelism [fðTÞ theories] take as a starting point theteleparallel equivalent of general relativity. We willsummarize here the basic elements needed to elaboratethe ideas of the present work, leaving the details for thereader. For a thorough introduction to fðTÞ gravity as wellas to its mathematical basis the reader can consult forinstance [35–39].The spirit of the equivalence between the Riemann and

Weitzenböck formulations of GR can be summarized in theequation

T ¼ −Rþ 2 e−1∂νðeTσσνÞ: (1)

On the left-hand side of Eq. (1) we have the so-calledWeitzenböck invariant

T ¼ SρμνTρμν; (2)

where Tρμν are the components (on a coordinate basis) of

the torsion 2-form Ta ¼ dea coming from the Weitzenböckconnection Γλ

νμ ¼ eλa∂νeaμ, and Sλμρ is defined according to

Sρμν ¼ 1

4ðTρ

μν − Tρμν þ Tρ

νμÞ þ1

2δρμTσ

σν − 1

2δρνTσ

σμ: (3)

Actually, T is the result of a very specific quadraticcombination of irreducible representations of the torsiontensor under the Lorentz group SOð1; 3Þ [40]. Equation (1)simply says that the Weitzenböck invariant T differs fromthe scalar curvature R in a total derivative; therefore, bothconceptual frameworks are totally equivalent at the time ofdescribing the dynamics of the gravitational field.fðTÞ gravity can be viewed as a natural extension of

Einstein gravity, and is ruled by the action in D spacetimedimensions

S ¼ 1

16πG

ZdDxe½fðTÞ þ Lmatter�; (4)

being e ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidetðgμνÞ

p. Of course GR is contained in (4) as

the particular case when fðTÞ ¼ T. The dynamical equa-tions in fðTÞ theories, for matter coupled to the metric inthe usual way, are

ðe−1∂μðeSaμνÞ þ eλaTρμλSρμνÞf0ðTÞþ

Saμν∂μðTÞf00ðTÞ − 1

4eνafðTÞ ¼ −4πGeλaTλ

ν; (5)

where the prime means derivative with respect to T and Tλν

is the energy-momentum tensor. It is worth insisting thatEqs. (5) are second-order differential equations for thevielbein components. This implies a genuine advantagecompared with other deformed gravitational schemes suchas, for instance, the popular fðRÞ gravities. This goodness,however, has consequences: the lack of local Lorentzinvariance. If we perform on a given solution (let us sayeb), a local Lorentz transformation ea⟶ea

0 ¼ Λa0b ðxÞeb,

we will find that ea0is not a solution of the field equations

(5). The reason for this lack of local invariance is simple:under a local transformation the scalar (2) changes accord-ing to the T → T 0 ¼ Tþ surface term. So this surface term,which is harmless when fðTÞ ¼ T, remains inside thefunction f, thus ruining the local invariance of the theory.This is because the theory picks up preferred referentialframes that constitute the autoparallel curves of the givenmanifold [41]. In other words, the field equations (5)determine the full components of the vielbein and not

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just those of the metric tensor related to the vielbein bymeans of

gμνðxÞ ¼ ηabeaμðxÞebνðxÞ: (6)

The preferred reference frames that solve the dynamicalequations (5) constitute what we shall call the properframes. These frames define the spacetime structure bymeans of a global set of basis covering the whole tangentbundle T ðMÞ, i.e., they parallelize the spacetime (seeSec. IV for details). From the infinite set of vielbeins givinga certain metric, the motion equations (5) choose thoseframes which act as well-defined (smooth), non-null fieldson the whole tangent bundle. Equation (6) just says thatthese smooth basis fields are everywhere orthonormal.An important fact sometimes omitted in the literature

concerns the coupling with the matter fields. If we assumethat the matter action depends only on the metric (and noton the whole vielbein), then the nonfermionic matter fieldsare unable to sense the lack of local Lorentz invariance byvirtue of the invariant character of expression (6) undersuch transformations. Then, as far as coupling to (non-spinning) matter is concerned, the lack of local invarianceshould not be problematic. The effects of the additionaldegrees of freedom that certainly exist in fðTÞ theories as aconsequence of the breaking of the Lorentz invariance (seethe analysis performed in [42]) must be found beyond theunspinning matter. Actually, it was found that on the flatFriedmann-Robertson-Walker (FRW) background with ascalar field, linear perturbation up to second order does notreveal any extra degree of freedom at all [37]. It is fair tosay, though, that the nature of the additional degrees offreedom remains unknown.

III. TOROIDAL COMPACTIFICATIONSIN D − 4 INTERNAL DIMENSIONS

In this section we will study a D-dimensional cosmo-logical model inside fðTÞ-gravity theory. We will considerthe simplest case where the additional internal dimensionsare compactified in a torus, and in the next section we willdiscuss other topologies for the extra dimensions. Theexternal spacetime is assumed to be spatially flat inaccordance with the current experimental evidence. Thus,the topology of our cosmological model is describedby R × R3 × S1 × � � � × S1, where S1 × � � � × S1 is theðD − 4Þ-dimensional torus. Under these considerationsthe metric is written as

ds2 ¼ dt2 − a20ðtÞðdx2 þ dy2 þ dz2Þ −XD−4

n¼1

anðtÞdX2n;

(7)

where a0ðtÞ denotes the scale factor of the external spaceand anðtÞ ðn ¼ 1;…; D − 4Þ the scale factors for the

internal dimensions Xn. Based on these observations weknow that the internal compact dimensions must be verysmall in order to be undetectable, so we are mainlyinterested in solutions where all the radii of the internaldimensions anðtÞ ðn ¼ 1;…; D − 4Þ become smaller ascosmic time evolves. To proceed, we first need to findthe correct parallelization of the spacetime under consid-eration. For the metric (7), and due to the fact that the circleS1 is trivially parallelizable [just take any smooth, non-nulltangent vector field in T ðS1Þ], a vielbein that parallelizesspacetime is a simple diagonal one

eaμ ¼ diagð1; a0ðtÞ; a0ðtÞ; a0ðtÞ; a1ðtÞ; :::; aD−4ðtÞÞ: (8)

Actually, we can think about the vielbein (8) as follows:unwrap the periodic coordinates Xn, and take the sub-manifold t ¼ constant. After rescaling the coordinatesaccording to

x → x0 ¼ a0x;

y → y0 ¼ a0y;

z → z0 ¼ a0z;

Xn → X0n ¼ anXn;

we just get D − 1 Euclidian space. For this space, it isclear that the autoparallel lines are just the straight lineswhich can be generated by the basis ∂i ði∶1; :::D − 1Þ, thedual cobasis of which is dxi. So, up to a time-dependentconformal factor, the frames describing the autoparallellines are dxi, and then the whole spacetime is parallelizedas it is indicated in (8).In order to obtain the motion equations, we shall work in

the comoving system, where the energy-momentum tensorfor a perfect fluid reads

Tμν ¼ diagðρ;−p0;−p0;−p0;−p1;…;−pD−4Þ: (9)

In this way, the energy conservation equation takes theform

ρ: þ�3H0 þ

XD−4

n¼1

Hn

�ρþ 3H0p0 þ

XD−4

n¼1

Hnpn ¼ 0; (10)

where a dot means derivative with respect to time andH0 ¼ a

:0=a0 andHn ¼ a

:n=an are the Hubble parameters of

the submanifolds with the scale factors a0ðtÞ and anðtÞ,respectively. We can now compute the Weitzenböck invari-ant (2), which is

T ¼ −6�H2

0 þH0

XD−4

n¼1

Hn þ1

3

XD−4

n¼1

HnHnþ1

�: (11)

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The initial value equation, i.e., the equation coming fromthe variation of the action with respect to the e00 componentof the vielbein results

f − 2Tf0 ¼ 16πGρ: (12)

On the other hand, the action variation with respect tothe spatial sector of the vielbein leads to the followingequations:

2f0�3H2

0 þ 2H:

0 þXD−4

n¼1

ðH: n þ 2H0Hn þH2nÞ − T

2

�

þ2f00 _T�2H0 þ

XD−4

n¼1

Hn

�þ fðTÞ ¼ −16πGp0;

(13)

2f0�6H2

0 þ 3H:

0 þXD−4

n¼1;n≠bðH: n þH0Hn þH2

nÞ − T2

�

þ2f00 _T�3H0 þ

XD−4

n¼1;n≠bHn

�þ fðTÞ ¼ −16πGpb:

(14)

Note that this last expression actually contains D − 4equations. The full system of equations to be solved isgiven by Eqs. (12–14) with T given by (11). They containthe conservation equation (10) self-consistently.Now we will investigate some simple limit cases, in

order to analyze the behavior of the scale factors. In all ofthis section, we will consider that the Hubble parameters ofthe extra dimensions are equal (H1 ¼ � � � ¼ HD−4 ≡H1s)and the content of matter in the spacetime is assumed to bedust matter, i.e., p0 ¼ pn ¼ 0. In this way, Eqs. (11), (13),and (14) are simplified to

T¼−6H20−6ðD−4ÞH0H1−ðD−4ÞðD−5ÞH2

1; (15)

2f0�3H2

0þ2 _H0þðD−4Þð _H1þ2H0H1þH21Þ−T

2

�þ2f00 _Tð2H0þðD−4ÞH1ÞþfðTÞ¼0; (16)

2f0�6H2

0 þ 3 _H0 þ ðD − 5Þð _H1 þH0H1 þH21Þ − T

2

�þ 2f00 _Tð3H0 þ ðD − 5ÞH1Þ þ fðTÞ ¼ 0. (17)

First, we will analyze the limit case when both Hubbleparameters H0 and H1 are constants; therefore, from

Eq. (15) we know that T must be a constant too, andthe above equations reduce to a more manageable form

2f0�3H2

0 þ ðD − 4Þð2H0H1 þH21Þ − T

2

�þ fðTÞ ¼ 0;

(18)

2f0�6H2

0 þ ðD − 5ÞðH0H1 þH21Þ − T

2

�þ fðTÞ ¼ 0.

(19)

Therefore, Eqs. (18) and (19) yield the following condition:

H1 ¼ − 6H0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD − 3Þ2 þ 12

p − ðD − 3Þ : (20)

Having imposed the constancy of H0 and H1, we will seeunder what conditions the initial value and energy con-servation equations are satisfied. In this case, the energyconservation equation can be written as

_ρþ ð3H0 þ ðD − 4ÞH1Þρ ¼ 0: (21)

Using the fact that H0 ¼ _a0ðtÞ=a0ðtÞ and H1 ¼_a1ðtÞ=a1ðtÞ, the energy density is integrated straightfor-wardly from the above expression, leading to

ρ ¼ Ca30ðtÞaD−4

1 ðtÞ ; (22)

where C is an integration constant. The initial valueequation (12) is satisfied if ρ is constant (due to the constancyof T), which implies that the term a30ðtÞaD−4

1 ðtÞ must beconstant. This condition actuallymeans that the total volumeof the higher-dimensional space remains constant in time,whichwill ensure that a dynamical contraction of the internaldimensions yields an expanding external space. FromEq. (21) we obtain

H0 ¼ − ðD − 4Þ3

H1: (23)

This equation together with Eq. (20) is only satisfied forD ¼ 5. However, in five dimensions we haveH0 ¼ −H1=3and T ¼ 12H2

0. Thus, by replacing in Eq. (19), weobtain fðTÞ ¼ 0. In this way, Eq. (12) yields ρ ¼ 0 andtherefore the constantC in (22) is null. Also, it is easy to seethat fordimensionshigher thanfiveandforp0 ¼ pn ¼ ρ ¼ 0and a constant H0, the only possible solution is given byH0 ¼ H1 ¼ 0 with fðTÞ ¼ 0. Therefore, we conclude thatthis compactification is not able to describe a de Sitterexpansion in vacuum.

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Another limit case worth mentioning is when theHubble parameters of the extra dimensions vanish(H1 ¼ � � � ¼ HD−4 ¼ 0), which means that the internalspace has a constant volume that should be very small.This represents a very simple model describing the asymp-totic evolution of the additional dimensions to a finalconstant volume. It is easy to see that no solution of suchtype exists in this context. In contrast, we will see in Sec. IVthat this kind of behavior is possible for the extradimensions in some spherical compactifications.Finally, let us briefly comment on some additional exact

solutions occurring in D ¼ 5. Taking ρ ¼ 0 and non-constant Hubble parameters in Eqs. (11), (13), and (14)we get

T ¼ −6ðH20 þH0H1Þ; (24)

2f0�3H2

0 þ 2 _H0 þ _H1 þ 2H0H1 þH21 − T

2

�þ 2f00 _Tð2H0 þH1Þ þ fðTÞ ¼ 0; (25)

2f0�6H2

0 þ 3 _H0 − T2

�þ 6f00 _TH0 þ fðTÞ ¼ 0. (26)

From Eq. (12), we obtain

f2f0

¼ T: (27)

Therefore, T is a constant that depends on fðTÞ [43], andby replacing it in Eq. (26), we get

_H0 þ 2H20 þ

T6¼ 0; H1 ¼ − T

6H0

−H0; (28)

where we have used (24). By this manner we candistinguish three cases that depend on the sign of T.(i) Case T < 0. Equation (28) tells us in this case that the

Hubble parameter H0 yields

H0 ¼1

2

ffiffiffiffiffiffijTj3

rtanh

� ffiffiffiffiffiffijTj3

rðtþ BÞ

�; (29)

where B is an integration constant. It is worth notingthat both scale factors, which are given by

a0ðtÞ ¼ αcosh1=2� ffiffiffiffiffiffi

jTj3

rðtþ BÞ

�; (30)

a1ðtÞ ¼ β sinh

� ffiffiffiffiffiffijTj3

rðtþ BÞ

�

× cosh−1=2� ffiffiffiffiffiffi

jTj3

rðtþ BÞ

�; (31)

where α and β are constant, increase in time. This case,thus, seems to be unprovided of physical significance.

(ii) Case T > 0. Now H0 is given by

H0 ¼ − 1

2

ffiffiffiffiT3

rtan

� ffiffiffiffiT3

rðtþ BÞ

�; (32)

and the scale factors by

a0ðtÞ ¼ α cos1=2� ffiffiffiffi

T3

rðtþ BÞ

�; (33)

a1ðtÞ ¼ β sin

� ffiffiffiffiT3

rðtþ BÞ

�cos−1=2

� ffiffiffiffiT3

rðtþ BÞ

�:

(34)

In order to be physically admissible, at least as a modelof the early stages of cosmic evolution, we must bind

the proper time to the interval − π2

ffiffiffi3T

q≤ t ≤ 0, having

taken B ¼ 0. In this way a0 is an increasing function

of time, while a1 decreases from infinite at t ¼ − π2

ffiffiffi3T

qto a null value at t ¼ 0.

(iii) Case T ¼ 0. Finally, when the scalar torsion vanishesthe scale factors are given by

a20ðtÞ ¼ 2tþ B; (35)

a21ðtÞ ¼1

2tþ B: (36)

This behavior for the scale factors shows a suppressionof the extra dimensions as the bulk expands linearlyin time, and then, constitutes a physical admissiblesimple model of the early universe.

Note that in the three cases, the solutions are independent ofthe specific form of the function fðTÞ, in the sense thatthe sole role of fðTÞ is to fix the constant value of T. Weactually included them here as simple examples of exactstates of the theory, and we hope that they might serve as aseed for more realistic solutions.

IV. SPHERICAL COMPACTIFICATIONSIN D − 4 INTERNAL DIMENSIONS

A. Statement of the problem

In theories relying on absolute parallelism, i.e., intheories where the whole vielbein components (not justthe metric tensor) are determined by the field equations, thestructure of the parallel vector fields describing the addi-tional compact dimensions is highly nontrivial. In thissection we investigate this issue further and discuss anumber of topics essential to understanding the nature of

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the additional compact dimensions when they are otherthan the simple toroidal compactifications previously dis-cussed. In particular, we shall now focus on extra compactdimensions with topology Sn, n > 1.Our task is to find cosmological solutions of the fðTÞ

field equations (5) with D − 4 spherically compactifiedextra dimensions, i.e., a set of D 1-forms eaðxÞ that solvethe equations and lead to the metric tensor

ds2 ¼ dt2 − a21ðtÞdΩ2ðjÞ − a22ðtÞdΩ2

ðkÞ − � � �− as20ðtÞðdx2 þ dy2 þ dz2Þ; (37)

where dΩ2ðjÞ refers to the line element corresponding to the

j sphere, so jþ kþ � � � ¼ D − 4. In order to do so, weneed first to propose a proper frame field for the topologyR × Sj × Sk × � � � × R3, i.e., D 1-forms eaðxÞ which turnEqs. (5) in a set of consistent equations for the unknown(time-dependent) scale factors a0, a1, a2, … . Once this isdone, it will remain to solve the equations and to obtain thefunctional form of the scale factors. In Sec. III we haveexplained how to do this for extra dimensions with toroidaltopology, where the proper frame field was given in (8).Now we want to discuss this issue when the extradimensions are spherically compactified.We start noting that a remarkable result due to Stiefel

states that every orientable three-dimensional manifold isalso parallelizable [44], thus, so it is S3. This means thatin every orientable three-manifold M3 there exist threeindependent smooth, non-null vector fields covering theentire tangent bundle T ðM3Þ. Additionally, Kervaire andMilnor [45] have shown that, apart from S1 and S3, the onlyother parallelizable sphere is S7. For our purposes thismeans that extra-dimension spherical compactificationsother than these will possess a rather complicated parallelvector field structure.In general, a D-dimensional manifold MD will be

parallelizable if a global basis exists in T ðMDÞ. Thisglobal basis of the tangent bundle constitutes a preferredreference frame which can be used for defining the spacestructure. This is so because in this last case we can define agiven spacetime as the pair ðT ðMDÞ; eaðxÞÞ instead ofðMD; gμνðxÞÞ. This preferred reference frame eaðxÞ and thesolution of the Eqs. (5), as we shall see, are closely related.Remaining ever conscious of the subtleties involved, and

in order to illustrate the problem, let us consider first twospherically compactified extra dimensions. In this case weneed to find a parallelization eaðxÞ for the spacetime havingthe metric

ds2 ¼ dt2 − a21ðtÞðdθ2 þ sin2 θdϕ2Þ− a20ðtÞðdx2 þ dy2 þ dz2Þ; (38)

where ðθ;ϕÞ are standard spherical coordinates on the two-sphere. It is intuitively obvious that the six-dimensional

manifold described by (38) is parallelizable. As a matterof fact, the spacetime topology is M3 × R3, whereM3 ≈ R × S2, so the full spacetime manifold can be writtenas a product of two three-dimensional, orientable (i.e.,parallelizable) manifolds. However, it is important torealize that the parallel vector fields of M3 will be highlynontrivial because S2 itself is not parallelizable due to thehairy ball theorem, which states that S2 (like all the evenspheres) does not admit a global basis of the tangent bundleT ðS2Þ (see, for instance [46]). In order to better understandthis point, Fig. 1 below shows two attempts of coveringT ðS2Þ with a global, smooth, non-null vector field. Nomatter the arrangement, the field will not be non-null orsmooth in at least one point of T ðS2Þ.Taking into account these facts, we can arrange the

components of the vielbein corresponding to the metric(38) in a 6 × 6 matrix eaμ, so schematically we have

eaμ ¼

0B@

M3 O

Oa0 0 0

0 a0 0

0 0 a0

1CA; (39)

FIG. 1. Two infructuous attempts of parallelizing S2. In the upperpanel the field is smooth but null at the poles. In the lower panel it isnon-null at the poles but instead it fails to be smooth there.

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where O is the 3 × 3 null matrix andM3 is the 3 × 3 matrixrepresenting the parallel triad field of M3. The main pointwe wish to emphasize is thatM3 cannot be diagonal. If thiswere so, it would mean that the global vector fields whichperform the parallelization of M3 are just e1 ¼ dt, e2 ¼a1ðtÞdθ; and e2 ¼ a1ðtÞ sin θ dϕ, but this is impossiblebecause in this case the fields e2 and e3 should be aparallelization of S2, which does not exist at all.Even more, it is clear thatM3 cannot have the block form

M3 ¼ 1 0 0

0

0 M2

!(40)

either, where M2 is some 2 × 2 matrix. In this case,we would have that M2 is related to the matrixa1ðtÞdiagðdθ; sin θdϕÞ by means of a local rotation.However, again, it is impossible to find a global basisfor T ðS2Þ. This means that the matrixM3 should be relatedto diagðdt; a1ðtÞdθ; a1ðtÞ sin θdϕÞ by a Lorentz boost, andtherefore the structure (40) cannot be correct. Actually, weshould mention that the proper (autoparallel) vector fieldsof the form (39) corresponding to the metric (38) areunknown at present, and the task of finding the paralleliza-tion of M3 remains as an open problem. Fortunately, aparallelization of the manifold (38) can be found in asomewhat different manner, and this will be matter for theparagraphs below. There we will find that the correct vectorfields have the structure

eaμ ¼ 1 0 0

0

0 M5

!; (41)

where M5 is the matrix representing the parallelization ofthe five-dimensional manifold S2 × R3.As in the six-dimensional case, if one considers D ¼ 7,

the problem can be fully understood and this will also beexplained in detail below. For extra dimensions higherthan three more patience is required. If we take fourextra dimensions, in addition to the trivial toroidal

compactifications of the last section, we are led to thetopologies S1 × S3, S2 × S2, or S4. With the exception ofthe first case, where the parallelization is provided by asimple extension of the procedure to be explained below forthe S3 compactification, the parallelization will be highlynontrivial because S2 and S4 do not admit a global basis.This problem turns out to be even more difficult as thedimension increases, because the number of possiblecompactifications increases as D − 4, the number of extradimensions.

B. Complete characterization of D ¼ 7

As a working example, let us now consider the variouspossibilities arising from the case D ¼ 7. The first of thesewas worked with in the last section, where toroidalcompactifications were extensively discussed. Now wewill consider the seven-dimensional metric

ds2 ¼ dt2 − a21ðtÞ½dψ2 þ sin2ψðdθ2 þ sin2θdϕ2Þ�− a20ðtÞðdx2 þ dy2 þ dz2Þ; (42)

where we have supposed three additional dimensionsspherically compactified. The manifold topology isR × S3 × R3, which is a product of parallelizable mani-folds. Following the scheme (39) we will now have

eaμ ¼

0BBBBBBBB@

1 0 0 0 0 0 0

0

0 M3 O0

0 a0 0 0

0 O 0 a0 0

0 0 0 a0

1CCCCCCCCA; (43)

where nowM3 is the matrix representing the parallelizationof S3. A global basis for T ðS3Þ has been obtained in [35],so we will just summarize the result here. For M3 we have

M3 ¼ a1ðtÞ

0B@ cðθÞ −sðψÞcðψÞsðθÞ −s2ðψÞs2ðθÞ

sðθÞcðϕÞ sðψÞ½sðψÞsðϕÞ þ cðψÞcðθÞcðϕÞ� sðψÞsðθÞ½sðψÞcðθÞcðϕÞ − cðψÞsðϕÞ�sðθÞsðϕÞ sðψÞ½cðψÞcðθÞsðϕÞ − sðψÞcðϕÞ� sðψÞsðθÞ½cðψÞcðϕÞ þ sðψÞcðθÞsðϕÞ�

1CA; (44)

where we have used s≡ sin and c≡ cos. We can just add here that M3 can be obtained from the naive triada1ðtÞdiagðdψ ; sin ψdθ; sin ψ sin θdϕÞ by means of a local Euler rotation (see [35] for details). It is a simple exercise tocheck that the frame (43) with M3 given by (44) gives rise to the metric (42).Having found the proper frame, we can now proceed to compute the motion equations. The vielbein (43) leads to the

invariant

T ¼ −6ðH20 þH2

1 þ 3H0H1 − a−21 Þ: (45)

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The initial value equation then results in

f þ 12f0ðH20 þH2

1 þ 3H0H1Þ ¼ 16πGρ: (46)

On the other hand, by varying with respect to the spatialsector of the vielbein, we get two equations:

2f0�7H2

0 þ 4H21 þ 9H0H1 þ 3 _H0 þ 2 _H1 − T

3

�þ 2f00 _Tð3H0 þ 2H1Þ þ f ¼ −16πGp1; (47)

2f0ð6H20 þ 9H2

1 þ 15H0H1 þ 2 _H0 þ 3 _H1Þþ2f00 _Tð2H0 þ 3H1Þ þ f ¼ −16πGp0: (48)

All the remaining equations are trivial or equal to theselast two. This is proof that the frame (43) is actually aparallelization of (42), because it leads to a consistent set ofmotion equations.It is interesting to compare this compactification with the

toroidal ones (in D ¼ 7) worked with in the previoussection. A remarkable property of the system (46–48) isthat it admits a de Sitter expansion for the bulk while theextra dimensions remain compactified with a constant scalefactor, and that this happens for a pure vacuum. Actually, ifwe set H0 as a constant and H1 ¼ ρ ¼ p1 ¼ p2 ¼ 0, thesystem becomes [note that T ¼ −6ðH2

0 − a−21 Þ is constantin this case]

f þ 12H20f

0 ¼ 0; (49)

2f0�7H2

0 − T3

�þ f ¼ 0; (50)

Eq. (48) being equal to (49). Inserting the expressionT ¼ −6ðH2

0 − a−21 Þ and the value of f0 from (49) in (50)we immediately get

H20 ¼

2

3a−21 ; (51)

which implies T ¼ 3H20. Note that fðTÞ ¼ T is not a

solution of Eqs. (49) and (50), showing that this kind ofsolution is absent in Einstein’s theory. The result (51) isactually valid only for smooth deformations of GR, i.e., forfðTÞ models of the form fðTÞ ¼ T þ βgðTÞ. Replacingthis form of fðTÞ in Eq. (49), we obtain a relationshipbetween H0, β; and the potential constants appearingin gðTÞ.Also, note that at this point the constant value of H0 (or

a1) remains undetermined. Nevertheless, given that theobservational evidence suggests that the size of the extradimensions, if non-null, should be very small at the present

time, Eq. (51) focuses attention on the ultraviolet regime.This enables us to interpret inflation as a vacuum-drivenexpansion given by the small compact extra dimensions.For this purpose let us invoke, as an example, the highenergy deformation of GR with the Born-Infeld structureworked in [3] and [4]. We thus have

fðTÞ ¼ −λ½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − 2λ−1T

p− 1�; (52)

where λ is the Born-Infeld constant [so, β ¼ −λ−1,gðTÞ ¼ T2=2þ λ2OðT=λÞ3]. In this manner we have thatEinstein’s gravity (in its absolute parallelism form) isrecovered when T=λ << 1. The role of Eq. (49) will beto link the undetermined parameter H0 (or a1) with λ.Actually, replacing the functional form (52) in (49) we getthe quadratic expression

ð3uþ 1Þ2 ¼ 1þ u; u ¼ −6λ−1H20; (53)

which leads to the solutions H0 ¼ 0 and H20 ¼ 5λ=54. Of

course, H0 ¼ 0 just says that Minkowski spacetime is avacuum solution of the theory. In turn, H2

0 ¼ 5λ=54, whichis very close toH2

0 ¼ λ=12, the value obtained in [4] for theinflationary era, represents a de Sitter accelerated expan-sion, the cause of which is the presence of the extradimensions.Let us go back now to the full characterization ofD ¼ 7.

The remaining topology for the three additional dimensionsis just S1 × S2, so the full metric looks in this case like

ds2 ¼ dt2 − a22ðtÞdΘ2 − a21ðtÞðdθ2 þ sin2θdϕ2Þ− a20ðtÞðdx2 þ dy2 þ dz2Þ: (54)

We proceed now to find the parallel fields for the geometry(54). For this task we focus on the embedding of thesubmanifold M3 ¼ S1 × S2 in the four-dimensional spaceS1 × R3. Installing coordinates ðΘ; X; Y; ZÞ in this mani-fold, we have that a parallelization of T �ðM3Þ reads

E1 ¼ a2ðtÞZdΘ − a1ðtÞðYdX − XdYÞ;E2 ¼ a2ðtÞYdΘþ a1ðtÞðZdX − XdZÞ;E3 ¼ a2ðtÞXdΘ − a1ðtÞðZdY − YdZÞ: (55)

This base turns out to be a global parallelization forS1 × S2 only if a1ðtÞ and a2ðtÞ are non-null, so a singularityin the Riemannian sense [i.e., when at least either a1ðtÞ ora2ðtÞ vanishes], actually represents a singularity in theparallelization process. In terms of the basis ðΘ; θ;ϕÞ,where X ¼ sin θ cos ϕ, Y ¼ sin θ sin ϕ; and Z ¼ cos θwe therefore have

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E1 ¼ a2ðtÞ cos θdΘþ a1ðtÞsin2θdϕ;E2 ¼ a2ðtÞ sin θ sin ϕ dΘ

þ a1ðtÞðcos ϕ dθ − sin θ sin ϕ cos θdϕÞ;E3 ¼ a2ðtÞ sin θ cos ϕdΘ

− a1ðtÞðsin ϕ dθ þ cos θ sin θ cos ϕ dϕÞ: (56)

Being S1 × S2 a three-dimensional manifold, we can onlyprovide a glimpse of the structure underlying the fields (55);in every point of the submanifold Θ ¼ constant, we have

~E1 ¼ −YdX þ XdY;

~E2 ¼ ZdX − XdZ;

~E3 ¼ −ZdY þ YdZ; (57)

where ~Ei ¼ Ei=a1ðtÞ. These fields, as viewed immersedin 3D-Euclidian space, look similar to the ones of Fig. 1 ofRef. [35], where the parallelization (44) is shown for ψ ¼π=2 (i.e., for the S3 sphere hyperequator). However, thisresemblance is partial, and it is important to bear in mindthat the fields (44) and (56) represent parallelizations of twogenuinely different topological structures, S3 and S1 × S2

respectively.In analogy with the analysis performed before, we can

now obtain the motion equations. First, it is necessary tocompute the scalar invariant, which reads

T ¼ −2ð3H20 þH2

1 þ 6H0H1 þ 2H1H2 þ 3H0H2 − a−21 Þ:(58)

Note that just the scale factor corresponding to S2 is presentin T, as must be the case. Moreover, the scalars (45) and(58) are different even when a0 ¼ a1, because theyrepresent different geometries. The full set of motionequations is

fþ4f0ðH21þ3H2H0þ3H2

0þ2H1H2þ6H1H0Þ¼16πGρ;

(59)

fþ2f0ð4H21þ2H1H2þ9H2

0þ12H1H0þ3H0H2

þ2 _H1þ3 _H0Þþ2f00ð2H1þ3H0Þ _T¼−16πGp2; (60)

f þ 2f0ð−a−21 þ 2H21 þH2

2 þ 9H20 þ 3H1H2

þ 9H1H0 þ 6H2H0 þ _H1 þ _H2 þ 3 _H0Þþ 2f00ðH1 þH2 þ 3H0ÞT

: ¼ −16πGp1; (61)

f þ 2f0ð4H21 þ 6H2

0 þH22 þ 4H1H2 þ 10H1H0

þ 5H2H0 þ 2 _H1 þ _H2 þ 2H:

0Þþ 2f00ð2H1 þH2 þ 2H0Þ _T ¼ −16πGp0: (62)

A simple exercise shows that vacuum solutions withH1 ¼ H2 ¼ 0 and a non-null constant H0 does not exist atall for this specific compactification. Imposing theseconditions on (59) and (60) we are led to the inconsistent(when H0 ≠ 0) equations

f þ 12f0H20 ¼ 0;

f þ 18f0H20 ¼ 0. (63)

So, as far as vacuum-driven inflation given by the smallcompact extra dimensions is concerned, we see that the S3

compactification is clearly favored.

C. Complete characterization of D ¼ 6

In this subsection we consider the remaining topology(S2) for the internal dimensions of the six-dimensionalmanifold. We are now ready to go back to the metric(38). The topology of this manifold is now described byR × S2 × R3, so we proceed to parallelize this submanifoldusing a similar method to that employed at the end of theprevious subsection. The key point is that we can unwrapthe periodic coordinate Θ in (56) and think about it as one ofthe bulk coordinates, let us say x. In this way, we obtain thesubmanifold M3 ¼ S2 × R. Again, embedding this inorder to obtain three-dimensional manifold in the four-dimensional space R3 × R and installing coordinatesðX; Y; Z; xÞ on it, we obtain that a parallelization ofT �ðM3Þ reads

E1 ¼ a0ðtÞ cos θdxþ a1ðtÞsin2θdϕ;E2 ¼ a0ðtÞ sin θ sin ϕdx

þ a1ðtÞðcos ϕdθ − sin θ sin ϕ cos θdϕÞ;E3 ¼ a0ðtÞ sin θ cos ϕdx

− a1ðtÞðsin ϕdθ þ cos θ sin θ cos ϕdϕÞ; (64)

where again, X ¼ sin θ cos ϕ, Y ¼ sin θ sin ϕ; and Z ¼cos θ was used. An observation is in order: nothing preventsthinking of the unwrappedΘ periodic coordinate now as the yor z coordinate of the bulk given the isotropy and homo-geneity of the latter. This actually means that we obtain threedifferent but equivalent sets of parallel vector fields for (38).With this vielbein in mind we can obtain the following

torsion invariant and field equations,

T ¼ −2ð−a−21 þH21 þ 6H0H1 þ 3H2

0Þ; (65)

f þ 4f0ðH21 þ 6H0H1 þ 3H2

0Þ ¼ 16πGρ; (66)

f þ 2f0ð6H20 þ 10H0H1 þ 4H2

1 þ 2 _H0 þ 2 _H1Þþ f00 _Tð2H0 þ 2H1Þ ¼ −16πGp0; (67)

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f þ 2f0ð−a−21 þ 9H20 þ 9H0H1 þ 2H2

1 þ 3H:

0 þH:

1Þþ f00 _Tð3H0 þH1Þ ¼ −16πGp1: (68)

Again, we are interested in solutions with ρ ¼ p0 ¼p1 ¼ H1 ¼ 0 and constant H0. From (66) we immediatelyget

f þ 12f0H20 ¼ 0; (69)

which is consistent with (67). This expression replacedin (68) leads to

H20 ¼

1

3a−21 :

This last equation is the analog of (51). Note that this valueof H0 yields T ¼ 0 by using (65), which turns out to beproblematic. Actually, every ultraviolet deformation lookslike fðTÞ ¼ T þOðT2Þ, so we have fð0Þ ¼ 0, f0ð0Þ ¼ 1,and then (69) is satisfied only when H0 ¼ 0. We cantherefore conclude that physically admissible models withthis topology do not contain H1 ¼ 0 and a non-nullconstant H0 as the solution.

V. CONCLUDING COMMENTS

This work was devoted to the study of spacetimes withFRW4 ×MD−4 topology in the context of the new gravi-tational schemes with absolute parallelism known as fðTÞtheories. We have assumed that FRW4 represents thespatially flat Friedmann-Robertson-Walker manifold cor-responding to the four-dimensional spacetime of standardcosmology, whileMD−4 refers to the ðD − 4Þ-dimensionalcompact extra dimensions constituting internal space. Onone hand we focused the analysis on ðD − 4Þ-dimensionalmanifolds consisting of ðD − 4Þ copies of the torus, and onthe other hand, several spherical compactifications wereconsidered.While obtaining the proper parallel vector fields for the

toroidal compactifications realized in Sec. III is trivial,[because the ðD − 4Þ torus TD−4 ¼ S1 × � � � × S1 is aproduct of trivial parallelizable manifolds], the correspond-ing characterization of the spherical topologies is highlynontrivial. Several aspects of this last issue were discussedin Sec. IV, where the problem was posed in detail. Then,we characterized the six- and seven-dimensional casesby finding the proper vielbeins for M2 ¼ S2 andM3 ¼ S1 × S2, M3 ¼ S3 respectively. These vielbeinsconstitute the startingpoint for anyfðTÞcosmologicalmodelofcompactextradimensions,becausetheyrepresent thebasisresponsible for the parallelization of the manifold underconsideration, i.e., they define the spacetime structure.In both sections, we discussed a number of simple exact

solutions of the field equations. One of the simple solu-tions explored throughout the different compactifications

considered in this work concerns the existence of a constantscale factor for the compact dimensions while the externalscale factor expands exponentially in time. This situationcorresponds to a vacuum-driven inflation powered by thepresence of extra dimensions. We showed that for five andsix spacetime dimensions, this kind of behavior does notexist at all. In turn, this problem finds a solution in D ¼ 7but not for any kind of compactification, just for S3. In thislast case we found that for smooth deformations of GR,the relation between H0 and a1 is given by equationH2

0 ¼ 23a−21 . Therefore, Hubble rates characterizing the

inflationary era find their hugeness in the exceedingly tinyvalue of a1. We do not expect this exact solution to describeall the details in the compactification process, but merelythe asymptotic stage of it. We think that a solution for theinternal dimensions of the form ain ¼ a1 þ αExpð−H0tÞ,with a constant α certainly do exist in theory, though it willprobably require the presence of matter fields and the use ofnumerical techniques.However, the existence of the exact solution in D ¼ 7

with S3 topology for the extra dimensions leads us to thinkthat a similar exact solution will exist only in D ¼ 11,where the seven additional dimensions will be compactifiedwith topology S7. Note that S7 is the only remainingparallelizable sphere, so vielbein fields with productstructure FRW4 × S7 will constitute a parallelization ofthe entire manifold. In this respect, we proved in Sec. IIIthat toroidal compactifications certainly do not lead to thiskind of solution in any spacetime dimension D. It will be amatter for future investigations to find the parallel vectorfields of FRW4 × S7 and those of the other admissible11-dimensional spherical topologies, and confirm (or dis-miss) this conjecture. The veracity of this statement maymake it possible to detect interesting links between thesefðTÞ deformed gravitational schemes and the low energylimit of M theory.Much more work is certainly necessary to fully under-

stand the peculiarities present in the parallelization processbehind fðTÞ theories. In particular, the characterization ofall the spherical topologies in a cosmological context for agiven spacetime dimension (with D > 7) remains entirelyopen. We would like to think that our work constitutes afirst and significant step towards that goal.

ACKNOWLEDGMENTS

F. F. is indebted to F. Canfora for many enlighteningdiscussions about extra dimensions in cosmology. F. F. issupported by CONICET and Universidad de Río Negro.Y. V. is supported by FONDECYT Grant No. 11121148,and by Dirección de Investigación y Desarrollo,Universidad de la Frontera, DIUFRO Grant No. DI11-0071. P. G. and Y. V. acknowledge the hospitality of theCentro Atómico Bariloche where part of this work wasundertaken.

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