Research Article Time-Dependent Toroidal Compactification...

Research ArticleTime-Dependent Toroidal Compactification Proposals and theBianchi Type I Model: Classical and Quantum Solutions

L. Toledo Sesma, J. Socorro, and O. Loaiza

Departamento de Fı́sica, DCI, Universidad de Guanajuato, Campus León, Loma del Bosque No. 103 Colonia Lomas del Campestre,Apartado Postal E-143, 37150 León, GTO, Mexico

Correspondence should be addressed to J. Socorro; [email protected]

Received 19 January 2016; Revised 22 April 2016; Accepted 5 May 2016

Academic Editor: Elias C. Vagenas

Copyright © 2016 L. Toledo Sesma et al.This is an open access article distributed under theCreativeCommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Thepublication of this article was funded by SCOAP3.

We construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalardilaton field on a time-dependent torus. This approach is applied to anisotropic cosmological Bianchi type I model for whichwe study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactificationprocess. Under this approach, we present an isotropization mechanism for the Bianchi I cosmological model through the analysisof the ratio between the anisotropic parameters and the volume of the Universe which in general keeps constant or runs into zerofor late times. We also find that the presence of extra dimensions in this model can accelerate the isotropization process dependingon the momenta moduli values. Finally, we present some solutions to the correspondingWheeler-DeWitt (WDW) equation in thecontext of standard quantum cosmology.

1. Introduction

The 2015 release of Planck data has provided a detailedmap of cosmic microwave background (CMB) temperatureand polarization allowing us to detect deviations from anisotropic early Universe [1]. The evidence given by these dataleads us to the possibility of considering that there is no exactisotropy, since there exist small anisotropy deviations of theCMB radiation and apparent large angle anomalies. In thatcontext, there have been recent attempts to fix constraintson such deviations by using the Bianchi anisotropic models[2]. The basic idea behind these models is to consider thepresent observational anisotropies and anomalies as imprintsof an early anisotropic phase on the CMB which in turncan be explained by the use of different Bianchi models. Inparticular, Bianchi I model seems to be related to large angleanomalies [2] (and references therein).

In a different context, attempts to understand diverseaspects of cosmology, as the presence of stable vacua andinflationary conditions, in the framework of supergravity andstring theory have been considered in the last years [3–10].One of the most interesting features emerging from these

types of models consists in the study of the consequencesof higher dimensional degrees of freedom on the cosmologyderived from four-dimensional effective theories [11, 12].The usual procedure for that is to consider compactificationon generalized manifolds, on which internal fluxes haveback-reacted, altering the smooth Calabi-Yau geometry andstabilizing all moduli [10].

The goal of the present work is to consider in a simplemodel some of the above two perspectives; that is, we willconsider the presence of extra dimensions in a Bianchi Imodel with the purpose of tracking down the influence ofmoduli fields in its isotropization. For that wewill consider analternative procedure concerning the role played by moduli.In particular we will not consider the presence of fluxes,as in string theory, in order to obtain a moduli-dependentscalar potential in the effective theory. Rather, we are goingto promote some of the moduli to time-dependent fields byconsidering the particular case of a ten-dimensional gravitycoupled to a time-dependent dilaton compactified on a 6-dimensional torus with a time-dependent Kähler modulus.With the purpose to study the influence of such fields, weare going to ignore the dynamics of the complex structure

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016, Article ID 6705021, 12 pageshttp://dx.doi.org/10.1155/2016/6705021

2 Advances in High Energy Physics

modulos (for instance, by assuming that it is already stabilizedby the presence of a string field in higher scales). This willallow us to construct classical effective models with twomoduli.

However, we are also interested in possible quantumaspects of our model. Quantum implications on cosmologyfrommore fundamental theories are expected due to differentobservations. For instance, it has been pointed out thatthe presence of extra dimensions leads to an interestingconnection to the ekpyrotic model [13], which generatedconsiderable activity [8, 9, 14]. The essential ingredient inthesemodels (see, for instance, [11]) is to consider an effectiveaction with a graviton and a massless scalar field, the dilaton,describing the evolution of the Universe, while incorporatingsome of the ideas of pre-big-bang proposal [15, 16] in thatthe evolution of the Universe began in the far past. Onthe other hand, it is well known that relativistic theoriesof gravity such as general relativity or string theories areinvariant under reparametrization of time. Quantization ofsuch theories presents a number of problems of principleknown as “the problem of time” [17, 18]. This problem ispresent in all systems whose classical version is invariantunder time reparametrization, leading to its absence at thequantum level. Therefore, the formal question involves howto handle the classical Hamiltonian constraint,H ≈ 0, in thequantum theory. Also, connected with the problem of time isthe “Hilbert space problem” [17, 18] referring to the not at allobvious selection of the inner product of states in quantumgravity and whether there is a need for such a structure at all.The above features, as it is well known, point out the necessityto construct a consistent theory of gravity at quantum level.

Analyses of effective four-dimensional cosmologies de-rived from M-theory and Type IIA string theory wereconsidered in [19–21] where string fluxes are related to thedynamical behavior of the solutions. A quantum descriptionof the model was studied in [22, 23] where a flat Friedmann-Robertson-Walker geometry was considered for the extendedfour-dimensional space-time, while a geometry given by 𝑆1 ×𝑇

6 was assumed for the internal seven-dimensional space;however, the dynamical fields are in the quintom scheme,since one of the fields has a negative energy. Under a similarperspective, we study the Hilbert space of quantum states ona Bianchi I geometry with two time-dependent scalar moduliderived from a ten-dimensional effective action containingthe dilaton and the Kähler parameter from a six-dimensionaltorus compactification. We find that, in this case, the wavefunction of this Universe is represented by two factors, onedepending onmoduli and the second one depending on grav-itational fields.This behavior is a property of all cosmologicalBianchi Class A models.

The work is organized in the following form. In Section 2we present the construction of our effective action by com-pactification on a time-dependent torus, while in Section 3we study its Lagrangian and Hamiltonian descriptions usingas toymodel the Bianchi type I cosmologicalmodel. Section 4is devoted to finding the corresponding classical solutionsfor few different cases involving the presence or absence ofmatter content and the cosmological constant term. Using

the classical solutions found in previous sections, we presentan isotropization mechanism for the Bianchi I cosmolog-ical model, through the analysis of the ratio between theanisotropic parameters and the volume of the Universe,showing that, in all the cases we have studied, its valuekeeps constant or runs into zero for late times. In Section 5we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of standard quantumcosmology and finally our conclusions are presented inSection 6.

2. Effective Model

We start from a ten-dimensional action coupled with adilaton (which is the bosonic component common to allsuperstring theories), which after dimensional reduction canbe interpreted as a Brans-Dicke like theory [24]. In thestring frame, the effective action depends on two spacetime-dependent scalar fields: the dilaton Φ(𝑥𝜇) and the Kählermodulus 𝜎(𝑥𝜇). For simplicity, in this work we will assumethat these fields only depend on time. The high-dimensional(effective) theory is therefore given by

𝑆 =

1

2𝜅2

10

∫𝑑

10𝑋√−̂𝐺𝑒

−2Φ[̂R

(10)

+ 4̂𝐺

𝑀𝑁

∇𝑀Φ∇

𝑁Φ]

+ ∫𝑑

10𝑋√−̂𝐺̂L

𝑚,

(1)

where all quantities �̂� refer to the string frame while the ten-dimensional metric is described by

𝑑𝑠

2=̂𝐺𝑀𝑁𝑑𝑋

𝑀𝑑𝑋

𝑁= �̂�

𝜇]𝑑𝑥𝜇𝑑𝑥

]+ ℎ

𝑚𝑛𝑑𝑦

𝑚𝑑𝑦

𝑛, (2)

where𝑀,𝑁, . . . are the indices of the ten-dimensional spaceand Greek indices 𝜇, ], . . . = 0, . . . , 3 and Latin indices𝑚, 𝑛, . . . = 4, . . . , 9 correspond to the external and internalspaces, respectively. We will assume that the six-dimensionalinternal space has the form of a torus with a metric given by

ℎ𝑚𝑛= 𝑒

−2𝜎(𝑡)𝛿𝑚𝑛, (3)

with 𝜎 being a real parameter.Dimensionally reducing the first term in (1) to four-

dimensions in the Einstein frame (see the Appendix fordetails) gives

𝑆4=

1

2𝜅2

4

∫𝑑

4𝑥√−𝑔 (R − 2𝑔

𝜇]∇𝜇𝜙∇]𝜙

− 96𝑔

𝜇]∇𝜇𝜎∇]𝜎 − 36𝑔

𝜇]∇𝜇𝜙∇]𝜎) ,

(4)

where 𝜙 = Φ + (1/2) ln(̂𝑉), with ̂𝑉 given by

̂𝑉 = 𝑒

6𝜎(𝑡)Vol (𝑋6) = ∫𝑑

6𝑦. (5)

By considering only a time-dependence on the moduli, onecan notice that, for the internal volume Vol(𝑋

6) to be small,

the modulus 𝜎(𝑡) should be a monotonic increasing function

Advances in High Energy Physics 3

on time (recall that 𝜎 is a real parameter), while ̂𝑉 is time andmoduli independent.

Now, concerning the second term in (1), we will requireproperly defining the ten-dimensional stress-energy tensor̂𝑇𝑀𝑁

. In the string frame it takes the form

̂𝑇𝑀𝑁

= (

̂𝑇𝜇] 0

0̂𝑇𝑚𝑛

) , (6)

where ̂𝑇𝜇] and ̂𝑇𝑚𝑛 denote the four- and six-dimensional

components of ̂𝑇𝑀𝑁

. Observe that we are not consideringmixing components of ̂𝑇

𝑀𝑁among internal and external

components although nonconstrained expressions for ̂𝑇𝑀𝑁

have been considered in [25]. In the Einstein frame the four-dimensional components are given by

𝑇𝜇] = 𝑒

2𝜙̂𝑇𝜇]. (7)

It is important to remark that there are some dilemmas aboutwhat is the best frame to describe the gravitational theory.Here in this work we have taken the Einstein frame. Usefulreferences to find a discussion about string and Einsteinframes and its relationship in the cosmological context are[26–30].

Now with expression (4) we proceed to build theLagrangian and the Hamiltonian of the theory at the classicalregime employing the anisotropic cosmological Bianchi typeI model. The moduli fields will satisfy the Klein-Gordon likeequation in the Einstein frame as an effective theory.

3. Classical Hamiltonian

In order to construct the classical Hamiltonian, we are goingto assume that the background of the extended space isdescribed by a cosmological Bianchi type I model. For that,let us recall here the canonical formulation in the ADMformalism of the diagonal Bianchi Class A cosmologicalmodels; the metric has the form

𝑑𝑠

2= −𝑁 (𝑡) 𝑑𝑡

2+ 𝑒

2Ω(𝑡)(𝑒

2𝛽(𝑡))

𝑖𝑗𝜔

𝑖𝜔

𝑗, (8)

where 𝑁(𝑡) is the lapse function, 𝛽𝑖𝑗(𝑡) is a 3 × 3 diagonal

matrix, 𝛽𝑖𝑗= diag(𝛽

++√3𝛽

−, 𝛽

+−√3𝛽

−, −2𝛽

+),Ω(𝑡) and 𝛽

±

are scalar functions, known as Misner variables, and 𝜔𝑖 areone-forms that characterize each cosmological Bianchi typemodel [31] and obey the form 𝑑𝜔𝑖 = (1/2)𝐶𝑖

𝑗𝑘𝜔

𝑗∧ 𝜔

𝑘, with𝐶

𝑖

𝑗𝑘the structure constants of the corresponding model. The

one-forms for the Bianchi type I model are𝜔1 = 𝑑𝑥,𝜔2 = 𝑑𝑦,and𝜔3 = 𝑑𝑧. So, the correspondingmetric of the Bianchi typeI in Misner’s parametrization has the form

𝑑𝑠

2

𝐼= −𝑁

2𝑑𝑡

2+ 𝑅

2

1𝑑𝑥

2+ 𝑅

2

2𝑑𝑦

2+ 𝑅

2

3𝑑𝑧

2, (9)

where {𝑅𝑖}

3

𝑖=1are the anisotropic radii and they are given by

𝑅1= 𝑒

Ω+𝛽++√3𝛽

−,

𝑅2= 𝑒

Ω+𝛽+−√3𝛽

−,

𝑅3= 𝑒

Ω−2𝛽+,

𝑉 = 𝑅1𝑅2𝑅3= 𝑒

3Ω,

(10)

where 𝑉 is the volume function of this model.The Lagrangian density, with a matter content given by

a barotropic perfect fluid and a cosmological term, has astructure, corresponding to an energy-momentum tensor ofperfect fluid [32–36]

𝑇𝜇] = (𝑝 + 𝜌) 𝑢𝜇𝑢] + 𝑝𝑔𝜇] (11)

that satisfies the conservation law ∇]𝑇𝜇]= 0. Taking the

equation of state 𝑝 = 𝛾𝜌 between the energy density andthe pressure of the comovil fluid, a solution is given by 𝜌 =𝐶𝛾𝑒

−3(1+𝛾)Ω with 𝐶𝛾the corresponding constant for different

values of 𝛾 related to the Universe evolution stage. Then, theLagrangian density reads

Lmatt = 16𝜋𝐺𝑁√−𝑔𝜌 + 2√−𝑔Λ

= 16𝑁𝜋𝐺𝑁𝐶𝛾𝑒

−3(1+𝛾)Ω+ 2𝑁Λ𝑒

3Ω,

(12)

while the Lagrangian that describes the fields dynamics isgiven by

L𝐼=

𝑒

3Ω

𝑁

[6̇Ω

2

− 6̇𝛽

2

+− 6

̇𝛽

2

−+ 96�̇�

2+ 36

̇𝜙�̇� + 2

̇𝜙

2

+ 16𝜋𝐺𝑁

2𝜌 + 2Λ𝑁

2] ,

(13)

using the standard definition of the momenta, Π𝑞𝜇 =

𝜕L/𝜕�̇�𝜇, where 𝑞𝜇 = (Ω, 𝛽+, 𝛽

−, 𝜙, 𝜎), and we obtain

ΠΩ=

12𝑒

3Ω̇Ω

𝑁

,̇Ω =

𝑁𝑒

−3ΩΠΩ

12

,

Π±= −

12𝑒

3Ω̇𝛽±

𝑁

,̇𝛽±= −

𝑁𝑒

−3ΩΠ𝛽±

12

,

Π𝜙=

𝑒

3Ω

𝑁

[36�̇� + 4̇𝜙] ,

̇𝜙 =

𝑁𝑒

−3Ω

44

[3Π𝜎− 16Π

𝜙] ,

Π𝜎=

𝑒

3Ω

𝑁

[192�̇� + 36̇𝜙] , �̇� =

𝑁𝑒

−3Ω

132

[9Π𝜙− Π

𝜎] ,

(14)

and, introducing them into the Lagrangian density, we obtainthe canonical Lagrangian asLcanonical = Π𝑞𝜇 �̇�

𝜇−𝑁H. When

we perform the variation of this canonical lagrangian withrespect to 𝑁, 𝛿Lcanonical/𝛿𝑁 = 0, implying the constraintH

𝐼= 0. In our model the only constraint corresponds to


Hamiltonian density, which is weakly zero. So, we obtain theHamiltonian density for this model:

H𝐼=

𝑒

−3Ω

24

[Π

2

Ω− Π

2

+− Π

2

−−

48

11

Π

2

𝜙+

18

11

Π𝜙Π𝜎

−

1

11

Π

2

𝜎− 384𝜋𝐺

𝑁𝜌𝛾𝑒

3(1−𝛾)Ω− 48Λ𝑒

6Ω] .

(15)

We introduce a new set of variables in the gravitational partgiven by 𝑒𝛽1+𝛽2+𝛽3 = 𝑒3Ω = 𝑉 which corresponds to thevolume of the Bianchi type I Universe, in similar way to theflat Friedmann-Robetson-Walker metric (FRW) with a scalefactor. This new set of variables depends on Misner variablesas

𝛽1= Ω + 𝛽

++√3𝛽

−,

𝛽2= Ω + 𝛽

+−√3𝛽

−,

𝛽3= Ω − 2𝛽

+,

(16)

from which the Lagrangian density (13) can be transformedas

L𝐼=

𝑒

𝛽1+𝛽2+𝛽3

𝑁

(2̇𝜙

2

+ 36̇𝜙�̇� + 96�̇�

2+ 2

̇𝛽1

̇𝛽2

+ 2̇𝛽1

̇𝛽3+ 2

̇𝛽3

̇𝛽2+ 2Λ𝑁

2

+ 16𝜋𝐺𝑁

2𝜌𝛾𝑒

−(1+𝛾)(𝛽1+𝛽2+𝛽3)) ,

(17)

and the Hamiltonian density reads

H𝐼=

1

8

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

2

1− Π

2

2− Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3−

16

11

Π

2

𝜙+

6

11

Π𝜙Π𝜎−

1

33

Π

2

𝜎

− 128𝜋𝐺𝜌𝛾𝑒

(1−𝛾)(𝛽1+𝛽2+𝛽3)− 16Λ𝑒

2(𝛽1+𝛽2+𝛽3)] .

(18)

So far, we have built the classical Hamiltonian density froma higher dimensional theory; this Hamiltonian contains abarotropic perfect fluid, that we have added explicitly. Thenext step is to analyze three different cases involving termsin the classical Hamiltonian (18) and find solutions to each ofthem in the classical regime.

3.1. Isotropization. The current observations of the cosmicbackground radiation set a very stringent limit to theanisotropy of the Universe [37]; therefore, it is importantto consider the anisotropy of the solutions. We will denotethrough all our study derivatives of functions 𝐹 with respectto 𝜏 = 𝑁𝑡 by 𝐹 (the following analysis is similar to thispresented recently in the K-essence theory, since the corre-sponding action is similar to this approach [38]). Recallingthe Friedmann like equation to the Hamiltonian density

Ω

2= 𝛽

2

++ 𝛽

2

−+ 16𝜎

2+

1

3

𝜙

2+ 6𝜙

𝜎

+

8

3

𝜋𝐺𝜇𝛾𝑒

−3(1+𝛾)Ω+

Λ

3

,

(19)

we can see that isotropization is achieved when the termswith 𝛽2

±go to zero or are negligible with respect to the other

terms in the differential equation.We find in the literature thecriteria for isotropization, among others, (𝛽2

++𝛽

2

−)/𝐻

2→ 0,

(𝛽

2

++𝛽

2

−)/𝜌 → 0, that are consistent with our above remark.

In the present case the comparison with the density shouldinclude the contribution of the scalar field. We define ananisotropic density 𝜌

𝑎that is proportional to the shear scalar,

𝜌𝑎= 𝛽

2

++ 𝛽

2

−, (20)

and will compare it with 𝜌𝛾, 𝜌

𝜙, and Ω2. From the Hamilton

analysis we now see thaṫΠ𝜙= 0 → Π

𝜙= 𝑝

𝜙= constant,

̇Π𝜎= 0 → Π

𝜎= 𝑝

𝜎= constant,

̇Π+= 0 → Π

+= 𝑝

+= constant,

̇Π−= 0 → Π

−= 𝑝

−= constant,

(21)

and the kinetic energy for the scalars fields 𝜙 and 𝜎 areproportional to 𝑒−6Ω. This can be seen from definitions ofmomenta associated with Lagrangian (13) and the aboveequations.Then, defining 𝜅

Ωas 𝜅2

Ω∼ 𝑝

2

++𝑝

2

−+(16/132

2)(𝑝

𝜎−

9𝑝𝜙)

2+(1/(3⋅44

2))(3𝑝

𝜎−16𝑝

𝜙)

2+(1/(22⋅44))(𝑝

𝜎−9𝑝

𝜙)(3𝑝

𝜎−

16𝑝𝜙), we have that

𝜌𝑎∼ 𝑒

−6Ω,

𝜌𝜙,𝜎∼ 𝑒

−6Ω,

Ω

2∼

Λ

3

+ 𝜅Ω

2𝑒

−6Ω+ 𝑏

𝛾𝑒

−3(1+𝛾)Ω,

𝑏𝛾=

8

3

𝜋𝐺𝑁𝜇𝛾.

(22)

With the use of these parameters, we find the following ratios:𝜌𝑎

𝜌𝜙,𝜎

∼ constant,

𝜌𝑎

𝜌𝛾

∼ 𝑒

3Ω(𝛾−1),

𝜌𝑎

Ω2∼

1

𝜅Ω

2+ (Λ/3) 𝑒

6Ω+ 𝑏

𝛾𝑒3(1−𝛾)Ω

.

(23)

We observe that for an expanding Universe the anisotropicdensity is dominated by the fluid density (with the exceptionof the stiff fluid) or by theΩ2 term and then at late times theisotropization is obtained since the above ratios tend to zero.

4. Case of Interest

In this section we present the classical solutions for theHamiltonian density of Lagrangian (13) we have previouslybuilt in terms of a new set of variables (16), focusing on threedifferent cases. We start our analysis on the vacuum case,and on the case with a cosmological term Λ. Finally we willanalyze the general case considering matter content and acosmological term.


4.1. VacuumCase. To analyze the vacuumcase, wewill take intheHamiltonian density (18) that 𝜌

𝛾= 0 andΛ = 0, obtaining

that

H𝐼vac=

1

8

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

2

1− Π

2

2− Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3−

16

11

Π

2

𝜙+

6

11

Π𝜙Π𝜎−

1

33

Π

2

𝜎] .

(24)

The Hamilton equations for the coordinates fields �̇�𝑖=

𝜕H/𝜕𝑃𝑖and the corresponding momenta ̇𝑃

𝑖= 𝜕H/𝜕𝑞

𝑖

become

̇Π𝑖= −H ≡ 0 ⇒ Π

𝑖= constant,

̇Π𝜙= 0 ⇒ Π

𝜙= constant,

̇Π𝜎= 0 ⇒ Π

𝜎= constant,

𝛽

1=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

1+ Π

2+ Π

3] ,

𝛽

2=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

2+ Π

1+ Π

3] ,

𝛽

3=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

3+ Π

1+ Π

2] ,

𝜙

= 𝜙

0𝑒

−(𝛽1+𝛽2+𝛽3),

𝜎

= 𝜎

0𝑒

−(𝛽1+𝛽2+𝛽3).

(25)

The gravitational momenta are constant by mean of theHamiltonian constraint, first line in the last equation. In thisway, the solution for the sum of 𝛽

𝑖functions becomes

𝛽1+ 𝛽

2+ 𝛽

3= ln [𝜖𝜏 + 𝑏

0] , 𝜖 = 𝑏

1+ 𝑏

2+ 𝑏

3, (26)

with 𝑏𝑖= Π

𝑖and 𝑏

0being an integration constant. Therefore,

Misner variables are expressed as

Ω = ln (𝜖𝜏 + 𝑏0)

1/3

,

𝛽+= ln (𝜖𝜏 + 𝑏

0)

(𝜖−3𝑏3)/6𝜖

,

𝛽−= ln (𝜖𝜏 + 𝑏

0)

(𝑏3+2𝑏1−𝜖)/2√3𝜖

,

(27)

while moduli fields are given by

𝜙 =

𝜙0

𝜖

ln (𝜖𝜏 + 𝑏0) ,

𝜎 =

𝜎0

𝜖

ln (𝜖𝜏 + 𝑏0) ,

(28)

where 𝜙0= Π

𝜙and 𝜎

0= Π

𝜎. Notice that, in this case,

the associated external volume given by 𝑒3Ω and the modulifields 𝜙 and 𝜎 are all logarithmically increasing functions ontime, implying for the latter that the internal volume shrinksin size for late times, as expected, while the four-dimensionalUniverse expands into an isotropic flat spacetime.Theparam-eter 𝜌

𝑎goes to zero at very late times independently of 𝑏

0.

4.2. Cosmological Term Λ. The corresponding Hamiltoniandensity becomes

H𝐼=

1

8

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

2

1− Π

2

2− Π

2

3+ 2Π

1Π2

+ 2Π1Π3+ 2Π

2Π3−

16

11

Π

2

𝜙+

6

11

Π𝜙Π𝜎−

1

33

Π

2

𝜎

− 16Λ𝑒

2(𝛽1+𝛽2+𝛽3)] ,

(29)

and corresponding Hamilton’s equations are

Π

1= Π

2= Π

3= 4Λ𝑒

𝛽1+𝛽2+𝛽3,

𝛽

1=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

1+ Π

2+ Π

3] ,

𝛽

2=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

2+ Π

1+ Π

3] ,

𝛽

3=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

3+ Π

1+ Π

2] ,

𝜙

= 𝜙

0𝑒

−(𝛽1+𝛽2+𝛽3),

𝜎

= 𝜎

0𝑒

−(𝛽1+𝛽2+𝛽3),

(30)

where the constants𝜙0and𝜎

0are the same as in previous case.

To solve the last systemof equationswewill take the followingansatz:

Π1= Π

2+ 𝛿

2= Π

3+ 𝛿

3, (31)

with 𝛿2and 𝛿

3constants. By substituting into Hamiltonian

(29) we find a differential equation for the momentaΠ1given

by

1

Λ

Π

2

1− 3Π

2

1+ ]Π

1+ 𝛿

1= 0, (32)

where the corresponding constants are

] = 2 (𝛿2+ 𝛿

3) ,

𝛿1(𝑞0) = (𝛿

2− 𝛿

3)

2

− 𝑞0,

(33)

with the constant (related to the momenta scalar field)

𝑞0=

16

11

Π

2

𝜙−

6

11

Π𝜙Π𝜎+

1

33

Π

2

𝜎. (34)

The solution for Π1is thus

Π1=

]6

+

1

6

√]2 + 12𝛿1cosh (√3Λ𝜏) . (35)

Hence, using the last result (35) in the rest of equations in (30)we obtain that

𝛽1+ 𝛽

2+ 𝛽

3= ln(1

8

√

]2 + 12𝛿1

3Λ

sinh (√3Λ𝜏)) , (36)


and the corresponding solutions to the moduli fields are

𝜙 =

8𝜙2

√]2 + 12𝛿1

ln(tanh(√3Λ

2

𝜏)) ,

𝜎 =

8𝜎2

√]2 + 12𝛿1

ln(tanh(√3Λ

2

𝜏)) .

(37)

With the last results we see that the solutions to the Misnervariables (Ω, 𝛽

±) are given by

Ω =

1

3

ln(18

√

]2 + 12𝛿1

3Λ

sinh (√3Λ𝜏)) ,

𝛽+=

2

3

(𝛿2− 2𝛿

3)

√]2 + 12𝛿1

ln tanh(√3Λ

2

𝜏) ,

𝛽−= −

2

√3

𝛿2

√]2 + 12𝛿1

ln tanh(√3Λ

2

𝜏) .

(38)

and the isotropization parameter 𝜌𝑎is given by

𝜌𝑎=

Λ

3

(𝛿2− 2𝛿

3)

2

+ 𝛿

2

2

]2 + 12 ((𝛿2− 𝛿

3)

2

− 𝑞0)

coth2 (√3Λ

2

𝜏) . (39)

Notice that the external four-dimensional volume expands astime 𝜏 runs and it is affectedwhether 𝑞

0is positive or negative.

This in turn determines how fast the moduli fields evolve inrelation to each other (i.e., whether Π

𝜙is larger or smaller

than Π𝜎). Therefore, isotropization is reached independently

of the values of 𝑞0but expansion of the Universe (and

shrinking of the internal volume) is affected by it.

4.3. Matter Content and Cosmological Term. For this case,the corresponding Hamiltonian density becomes (18). So,using the Hamilton equation, we can see that the momentaassociated with the scalars fields 𝜙 and 𝜎 are constants, andwe will label these constants by 𝜙

1and 𝜎

1, respectively:

Π

1= Π

2= Π

3

= 4Λ𝑒

𝛽1+𝛽2+𝛽3+ 16𝜋𝐺 (1 − 𝛾) 𝜌

𝛾𝑒

−𝛾(𝛽1+𝛽2+𝛽3),

𝛽

1=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

1+ Π

2+ Π

3] ,

𝛽

2=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

2+ Π

1+ Π

3] ,

𝛽

3=

1

4

𝑒

−(𝛽1+𝛽2+𝛽3)[−Π

3+ Π

1+ Π

2] ,

𝜙

= 𝜙

2𝑒

−(𝛽1+𝛽2+𝛽3),

𝜎

= 𝜎

2𝑒

−(𝛽1+𝛽2+𝛽3).

(40)

Follow the same steps as in Section 4.2, we see that it ispossible to relate the momenta associated with 𝛽

1, 𝛽

2, and 𝛽

3

in the following way:

Π1= Π

2+ 𝑘

1= Π

3+ 𝑘

2. (41)

Also, the differential equation for fields 𝜙 and 𝜎 can bereduced to quadrature as

𝜙 = 𝜙2∫ 𝑒

−(𝛽1+𝛽2+𝛽3)𝑑𝜏,

𝜎 = 𝜎2∫ 𝑒

−(𝛽1+𝛽2+𝛽3)𝑑𝜏.

(42)

We proceed to consider specific values for 𝛾.

4.3.1. 𝛾 = ±1 Cases. Let us introduce the generic parameter

𝜆 =

{

{

{

4Λ, 𝛾 = 1

4Λ + 32𝜋𝐺𝜌−1, 𝛾 = −1

(43)

for whichΠ1= 𝜆𝑒

𝛽1+𝛽2+𝛽3 . Substituting this intoHamiltonian

(18) we find the differential equation for the momenta Π1as

4

𝜆

Π

2

1− 3Π

2

1+ ]Π

1+ 𝛿

1= 0, (44)

with the solution given by Π1is

Π1=

]6

+

1

6

√]2 + 12𝛿1cosh(

√3𝜆

2

𝜏) . (45)

On the other hand, using the last result (45) we obtain fromHamilton equations that

𝛽1+ 𝛽

2+ 𝛽

3= ln(1

4

√

𝛿3+ 12𝛿

1

3𝜆

sinh(√3𝜆

2

𝜏)) . (46)

The corresponding solutions to the moduli fields (42) can befound as

𝜙 =

8𝜙2

√]2 + 12𝛿1

ln tanh(√3𝜆

4

𝜏) ,

𝜎 =

8𝜎2

√]2 + 12𝛿1

ln tanh(√3𝜆

4

𝜏) .

(47)


The solutions to Misner variables (16), (Ω, 𝛽±), are given by

Ω =

1

3

ln(14

√

𝛿

2+ 12𝛿

1

3𝜆

sinh(√3𝜆

2

𝜏)) ,

𝛽+=

2

3

(𝛿2− 2𝛿

3)

√]2 + 12𝛿1

ln tanh(√3𝜆

2

𝜏)

+

1

6

ln(14

√

]2 + 12𝛿1

3𝜆

) ,

𝛽−= −

2

√3

𝛿2

√]2 + 12𝛿1

ln tanh(√3𝜆

2

𝜏)

−

1

2√3

ln(14

√

]2 + 12𝛿1

3𝜆

) .

(48)

It is remarkable that (42) for the fields 𝜙 and 𝜎 aremaintainedfor all Bianchi Class A models, and in particular, when weuse the gauge𝑁 = 𝑒𝛽1+𝛽2+𝛽3 , the solutions for these fields areindependent of the cosmological models, whose solutions are𝜙 = (𝜙

0/8)𝑡 and 𝜎 = (𝜎

0/8)𝑡.

After studying the 𝛾 = 0 case we will comment on thephysical significance of these solutions.

4.3.2. 𝛾 = 0 Case. TheHamilton equations are

Π

1= Π

2= Π

3= 4Λ𝑒

𝛽1+𝛽2+𝛽3+ 𝛼

0, 𝛼

0= 16𝜋𝐺𝜌

0

Π

𝜙= 0, Π

𝜙= 𝜙

1= 𝑐𝑡𝑒,

(49)

from which the momenta Π1fulfil the equation

1

Λ

Π

2

1− 3Π

2

1+ 𝜇Π

1+ 𝜉

1= 0, (50)

where

𝜉1= 𝛿

1−

𝛼

2

0

Λ

.(51)

The solution is given by

Π1=

]6

+

1

6

√]2 + 12𝜉1cosh (√3Λ𝜏) . (52)

Using this result in the corresponding Hamilton equations,we obtain that

𝛽1+ 𝛽

2+ 𝛽

3= ln(1

8

√

]2 + 12𝜉1

3Λ

sinh (√3Λ𝜏)) , (53)

and the corresponding solutions to the dilaton field 𝜙 and themoduli field 𝜎 are given by

𝜙 =

8𝜙2

√]2 + 12𝜉1

ln tanh(√3Λ

2

𝜏) ,

𝜎 =

8𝜎2

√]2 + 12𝜉1

ln tanh(√3Λ

2

𝜏) .

(54)

So, the solutions to the Misner variables (Ω, 𝛽±) are

Ω =

1

3

ln(18

√

]2 + 12𝜉1

3Λ

sinh (√3Λ𝜏)) ,

𝛽+=

2

3

𝛿2− 2𝛿

3

√]2 + 12𝜉1

ln tanh(√3Λ

2

𝜏) ,

𝛽−= −

2

√3

𝛿2

√]2 + 12𝜉1

ln tanh(√3Λ

2

𝜏) .

(55)

For both solutions, 𝛾 = 0, ±1, we also observe that, forpositive (negative) 𝑞

0, the Universe expands slower (faster)

than in the absence of extra dimensions while the internalspace shrinks faster (slower) for positive (negative) 𝑞

0. In this

case, isotropization parameters are also accordingly affectedby 𝑞

0, through relations (34).

So far, our analysis has been completely classic. Now, inthe next section we deal with the quantum scheme and weare going to solve the WDW equation in standard quantumcosmology.

5. Quantum Scheme

Solutions to the Wheeler-DeWitt (WDW) equation dealingwith different problems have been extensively used in theliterature. For example, the important quest of finding atypical wave function for the Universe was nicely addressedin [39], while in [40] there appears an excellent summaryconcerning the problem of how a Universe emerged froma big bang singularity, whihc cannot longer be neglectedin the GUT epoch. On the other hand, the best candidatesfor quantum solutions become those that have a dampingbehavior with respect to the scale factor, represented in ourmodel with the Ω parameter, in the sense that we obtain agood classical solution using the WKB approximation in anyscenario in the evolution of our Universe [41]. The WDWequation for thismodel is achieved by replacing themomentaΠ𝑞𝜇 = −𝑖𝜕

𝑞𝜇 , associated with the Misner variables (Ω, 𝛽

+, 𝛽

−)

and the moduli fields (𝜙, 𝜎) in Hamiltonian (15). The factor𝑒

−3Ω may be factor ordered with ̂ΠΩin many ways. Hartle

and Hawking [41] have suggested what might be called asemigeneral factor ordering which in this case would order𝑒

−3Ω̂Π

2

Ωas

−𝑒

−(3−𝑄)Ω𝜕Ω𝑒

−𝑄Ω𝜕Ω= −𝑒

−3Ω𝜕

2

Ω+ 𝑄𝑒

−3Ω𝜕Ω, (56)

where 𝑄 is any real constant that measure the ambiguity inthe factor ordering in the variable Ω and the correspondingmomenta. We will assume in the following this factor order-ing for the Wheeler-DeWitt equation, which becomes

◻Ψ + 𝑄

𝜕Ψ

𝜕Ω

−

48

11

𝜕

2Ψ

𝜕𝜙2−

1

11

𝜕

2Ψ

𝜕𝜎2+

18

11

𝜕

2Ψ

𝜕𝜙𝜕𝜎

− [𝑏𝛾𝑒

3(1−𝛾)Ω+ 48Λ𝑒

6Ω]Ψ = 0,

(57)


where ◻ is the three-dimensional d’Lambertian in the ℓ𝜇 =(Ω, 𝛽

+, 𝛽

−) coordinates, with signature (− + +). On the other

hand, we could interpret the WDW equation (57) as a time-reparametrization invariance of the wave function Ψ. At aglance, we can see that the WDW equation is static; this canbe understood as the problem of time in standard quantumcosmology. We can avoid this problem by measuring thephysical time with respect a kind of time variable anchoredwithin the system, whichmeans that we could understand theWDW equation as a correlation between the physical timeand a fictitious time [42, 43]. When we introduce the ansatz

Ψ = Φ (𝜙, 𝜎) 𝜓 (Ω, 𝛽±) , (58)

in (57), we obtain the general set of differential equations(under the assumed factor ordering):

◻𝜓 + 𝑄

𝜕𝜓

𝜕Ω

− [𝑏𝛾𝑒

3(1−𝛾)Ω+ 48Λ𝑒

6Ω−

𝜇

2

5

]𝜓 = 0, (59)

48

11

𝜕

2Φ

𝜕𝜙2+

1

11

𝜕

2Φ

𝜕𝜎2−

18

11

𝜕

2Φ

𝜕𝜙𝜕𝜎

+ 𝜇

2Φ = 0, (60)

where we choose the separation constant 𝜇2/5 for conve-nience to reduce the second equation. The solution to thehyperbolic partial differential equation (60) is given by

Φ(𝜙, 𝜎) = 𝐶1sin (𝐶

3𝜙 + 𝐶

4𝜎 + 𝐶

5)

+ 𝐶2cos (𝐶

3𝜙 + 𝐶

4𝜎 + 𝐶

5) ,

(61)

where {𝐶𝑖}

5

𝑖=1are integration constants and they are in

terms of 𝜇. We claim that this solution is the same for allBianchi Class A cosmological models, because the Hamil-tonian operator in (57) can be written in separated way aŝ𝐻(Ω, 𝛽

±, 𝜙, 𝜎)Ψ =

̂𝐻𝑔(Ω, 𝛽

±)Ψ +

̂𝐻𝑚(𝜙, 𝜎)Ψ = 0, where

̂𝐻𝑔y ̂𝐻

𝑚represents the Hamiltonian to gravitational sector

and the moduli fields, respectively. To solve (59), we now set𝜓(Ω, 𝛽

±) = A(Ω)B

1(𝛽

+)B

2(𝛽

−), obtaining the following set

of ordinary differential equations:

𝑑

2A

𝑑Ω2− 𝑄

𝑑A

𝑑Ω

+ [𝑏𝛾𝑒

−3(𝛾−1)Ω+ 48Λ𝑒

6Ω+ 𝜎

2]A = 0,

𝑑

2B1

𝑑𝛽2

+

+ 𝑎

2

2B

1= 0 ⇒

B1= 𝜂

1𝑒

𝑖𝑎2𝛽++ 𝜂

2𝑒

−𝑖𝑎2𝛽+,

𝑑

2B2

𝑑𝛽2

−

+ 𝑎

2

3B

2= 0 ⇒

B2= 𝑐

1𝑒

𝑖𝑎3𝛽−+ 𝑐

2𝑒

−𝑖𝑎3𝛽−,

(62)

where 𝜎2 = 𝑎21+ 𝑎

2

2+ 𝑎

2

3and 𝑐

𝑖and 𝜂

𝑖are constants. We now

focus on the Ω dependent part of the WDW equation. Wesolve this equation when Λ = 0 and Λ ̸= 0.

(1) For the casewith null cosmological constant and 𝛾 ̸= 1,

𝑑

2A

𝑑Ω2− 𝑄

𝑑A

𝑑Ω

+ [𝑏𝛾𝑒

−3(1−𝛾)Ω+ 𝜎

2]A = 0, (63)

and by using the change of variable,

𝑧 =

√𝑏𝛾

𝑝

𝑒

−(3/2)(𝛾−1)Ω,

(64)

where 𝑝 is a parameter to be determined, we have

𝑑A

𝑑Ω

=

𝑑A

𝑑𝑧

𝑑𝑧

𝑑Ω

= −

3

2

(𝛾 − 1) 𝑧

𝑑A

𝑑𝑧

,

𝑑

2A

𝑑Ω2=

9

4

(𝛾 − 1)

2

𝑧

2 𝑑2A

𝑑𝑧2+

9

4

(𝛾 − 1)

2

𝑧

𝑑A

𝑑𝑧

.

(65)

Hence, we arrive at the equation

9

4

(𝛾 − 1)

2

𝑧

2 𝑑2A

𝑑𝑧2+

9

4

(𝛾 − 1)

2

𝑧

𝑑A

𝑑𝑧

−

3

2

𝑄 (𝛾 − 1) 𝑧

𝑑A

𝑑𝑧

+ [𝑝

2𝑧

2+ 𝜎

2] = 0,

(66)

where we have assumed thatA is of the form [44]

A = 𝑧𝑞𝑄Φ (𝑧) , (67)

with 𝑞 being yet to be determined. We thus get, aftersubstituting in (66),

9

4

(𝛾 − 1)

2

𝑧

𝑞𝑄[𝑧

2 𝑑2Φ

𝑑𝑧2+ 𝑧(1 + 2𝑞𝑄

+

2

3

𝑄

𝛾 − 1

)

𝑑Φ

𝑑𝑧

+ (

4𝑝

2

9 (𝛾 − 1)

2𝑧

2

+ 𝑄

2{𝑞

2+

2

3

𝑞

𝛾 − 1

} +

4𝜎

2

9 (𝛾 − 1)

2)Φ] = 0,

(68)

which can be written as

𝑧

2 𝑑2Φ

𝑑𝑧2+ 𝑧

𝑑Φ

𝑑𝑧

+ [𝑧

2−

1

9 (𝛾 − 1)

2(𝑄

2− 4𝜎

2)]Φ

= 0,

(69)

which is the Bessel differential equation for the function Φwhen 𝑝 and 𝑞 are fixed to

𝑞 = −

1

3 (𝛾 − 1)

,

𝑝 =

3

2

𝛾 − 1

,

(70)

which in turn means that transformations (64) and (67) are

𝑧 =

2√𝑏𝛾

3

𝛾 − 1

𝑒

−(3/2)(𝛾−1)Ω,

A = 𝑧−(𝑄/3(𝛾−1))Φ(𝑧)

.

(71)


Hence, the solution for (63) is of the form

A𝛾= 𝑐

𝛾(

2√𝑏𝛾

3

𝛾 − 1

𝑒

−(3/2)(𝛾−1)Ω)

−𝑄/3(𝛾−1)

⋅ 𝑍𝑖](

2√𝑏𝛾

3

𝛾 − 1

𝑒

−(3/2)(𝛾−1)Ω),

(72)

with

] = ±√1

9 (𝛾 − 1)

2(4𝜎

2− 𝑄

2), (73)

where𝑍𝑖] = 𝐽𝑖] is the ordinaryBessel functionwith imaginary

order. For the particular case when the factor ordering is zero,we can easily construct a wave packet [41, 45, 46] using theidentity:

∫

∞

−∞

sech(𝜋𝜂

2

) 𝐽𝑖𝜂(𝑧) 𝑑𝜂 = 2 sin (𝑧) , (74)

so, the total wave function becomes

Ψ (Ω, 𝛽±, 𝜙, 𝜎) = Φ (𝜙, 𝜎)

⋅ sin(2√𝑏𝛾

3

𝛾 − 1

𝑒

−(3/2)(𝛾−1)Ω)

⋅ [𝜂1𝑒


2𝑒

−𝑖𝑎2𝛽+]

⋅ [𝑐1𝑒


2𝑒

−𝑖𝑎3𝛽−] .

(75)

Notice that the influence of extra dimensions through thepresence of the moduli 𝜙 and 𝜎 appears in the solution inthe amplitude of the wave function.We will comment on thislater on.

(2) Case with null cosmological constant and 𝛾 = 1. Forthis case we have

𝑑

2A1

𝑑Ω2− 𝑄

𝑑A1

𝑑Ω

+ 𝜎

2

1A1= 0, 𝜎

2

1= 𝑏

1+ 𝜎

2,

(76)

whose solution is

A1= 𝐴

1𝑒

((𝑄+√𝑄2+4𝜎2

1)/2)Ω

+ 𝐴2𝑒

((𝑄−√𝑄2+4𝜎2

1)/2)Ω

.(77)

So, the total wave function becomes

Ψ (Ω, 𝛽±, 𝜙, 𝜎) = Φ (𝜙, 𝜎)

⋅ [𝐴1𝑒

((𝑄+√𝑄2+4𝜎2

1)/2)Ω

+ 𝐴2𝑒

((𝑄−√𝑄2+4𝜎2

1)/2)Ω

]

⋅ [𝜂1𝑒


2𝑒

−𝑖𝑎2𝛽+] [𝑐

1𝑒


2𝑒

−𝑖𝑎3𝛽−] .

(78)

(3) If we include the cosmological constant term for thisparticular case, we have [44]

A = 𝑒(𝑄/2)Ω

𝑍] (4√Λ

3

𝑒

3Ω) , ] = ±

1

6

√𝑄2+ 4𝜎

2

1, (79)

where Λ > 0 in order to have ordinary Bessel functionsas solutions; in other case, we will have the modified Besselfunction. When the factor ordering parameter𝑄 equals zero,we have the same generic Bessel functions as solutions,having the imaginary order ] = ±𝑖√𝜎2

1/3.

(4) 𝛾 = −1 and Λ ̸= 0 and any factor ordering 𝑄:

𝑑

2A−1

𝑑Ω2− 𝑄

𝑑A−1

𝑑Ω

+ [𝑑−1𝑒

6Ω− 𝜎

2]A

−1= 0,

𝑑−1= 48𝜇

−1+ 48Λ

(80)

with the solution

A−1= (

√𝑑−1

3

𝑒

3Ω)

𝑄/6

𝑍] (√𝑑

−1

3

𝑒

3Ω) ,

] = ±1

6

√𝑄2+ 4𝜎

2,

(81)

where𝑍] is a generic Bessel function. When 𝑏−1 > 0, we haveordinary Bessel functions; in other case, we have modifiedBessel functions.

(5) 𝛾 = 0, Λ ̸= 0, and factor ordering 𝑄 = 0

𝑑

2A0

𝑑Ω2+ (48Λ𝑒

6Ω+ 𝑏

0𝑒

3Ω+ 𝜎

2)A

0= 0, 𝑏

0= 48𝜇

0(82)

with the solution

A0= 𝑒

−3Ω/2[𝐷

1𝑀

−𝑖𝑏0/24√3Λ,−𝑖𝜎/3

(

8𝑖𝑒

3Ω√Λ

√3

)

+ 𝐷2𝑊−𝑖𝑏0/24√3Λ,−𝑖𝜎/3

(

8𝑖𝑒

3Ω√Λ

√3

)] ,

(83)

where𝑀𝑘,𝑝

And𝑊𝑘,𝑝

are Whittaker functions and𝐷𝑖are in-

tegration constants.

6. Final Remarks

In this work we have explored a compactification of a ten-dimensional gravity theory coupled with a time-dependentdilaton into a time-dependent six-dimensional torus. Theeffective theory which emerges through this process resem-bles the Einstein frame to that described by the BianchiI model. By incorporating the barotropic matter and cos-mological content and by using the analytical procedureof Hamilton equation of classical mechanics, in appropri-ate coordinates, we found the classical solution for theanisotropic Bianchi type I cosmologicalmodels. In particular,the Bianchi type I is completely solved without using aparticular gauge. With these solutions we can validate ourqualitative analysis on isotropization of the cosmologicalmodel, implying that the volume becomes larger in thecorresponding time evolution.

We find that, for all cases, the Universe expansion isgathered as time runs while the internal space shrinks.Qualitatively this model shows us that extra dimensions are


forced to decrease its volume for an expanding Universe.Also we notice that the presence of extra dimensions affectshow fast the Universe (with matter) expands through thepresence of a constant related to the moduli momenta. Thisis not unexpected since we are not considering a potentialfor the moduli, which implies that they are not stabilizedand consequently an effective model should only take intoaccount their constant momenta.We observe that isotropiza-tion is not affected in the cases without matter, but in thematter presence, isotropization can be favored or retardedaccording to how fast the moduli evolve with respect to eachother.

It could be interesting to study other types of matterin this context, beyond the barotropic matter. For instance,the Chaplygin gas, with a proper time, characteristic to thismatter, leads to the presence of singularities types I, II,III, and IV (generalizations to these models are presentedin [47–49]) which also appear in the phantom scenario todark/energy matter. In our case, since the matter we areconsidering is barotropic, initial singularities of these typesdo not emerge from our analysis, implying also that phantomfields are absent. It is important to remark that this model isa very simplified one in the sense that we do not considerthe presence of moduli-dependent matter and we do notanalyze under which conditions inflation is present or howit starts. We plan to study this important task in a differentwork.

Concerning the quantum schemewe can observe that thisanisotropic model is completely integrable without employ-ing numerical methods, similar solutions to partial differ-ential equation into the gravitational variables have beenfound in [50], and we obtain that the solutions in the modulifields are the same for all Bianchi Class A cosmologicalmodels, because the Hamiltonian operator in (57) can bewritten in separated way as ̂𝐻(Ω, 𝛽

±, 𝜙, 𝜎)Ψ =

̂𝐻𝑔(Ω, 𝛽

±)Ψ +

̂𝐻𝑚(𝜙, 𝜎)Ψ = 0, where ̂𝐻

𝑔y ̂𝐻

𝑚represents the Hamiltonian

to gravitational sector and themoduli fields, respectively, withthe full wave function given byΨ = Φ(𝜙, 𝜎)Θ(Ω, 𝛽

±). In order

to have the best candidates for quantum solutions becomethose that have a damping behavior with respect to the scalefactor, represented in ourmodelwithΩparameter, in thiswaywe will drop in the full solution the modified Bessel function𝐼](𝑧) or Bessel function 𝑌](𝑧), with ] being the order of thesefunctions.

We can observe that the quantum solution in the Ωsector is similar to the corresponding FRW cosmologicalmodel, found in different schemes [12, 22, 23, 32, 43]. Also,similar analysis based on a Bianchi type II model can befound in recent published paper by two authors of this work[51].

The presence of extra dimensions could have a strongerinfluence on the isotropization process by having a moduli-dependent potential, which can be gathered by turning onextra fields in the internal space, called fluxes. It will beinteresting to consider a more complete compactificationprocess inwhich allmoduli are considered as time-dependentfields as well as time-dependent fluxes. We leave theseimportant analyses for future work.

Appendix

Dimensional Reduction

With the purpose to be consistent, here we present the mainideas to dimensionally reduce action (1). By the use of theconformal transformation

̂𝐺𝑀𝑁

= 𝑒

Φ/2𝐺

𝐸

𝑀𝑁, (A.1)

action (1) can be written as

𝑆 =

1

2𝜅2

10

∫𝑑

10𝑋√−̂𝐺(𝑒

Φ/2̂R + 4𝐺

𝐸𝑀𝑁∇𝑀Φ∇

𝑁Φ)

+ ∫𝑑

10𝑋√−𝐺

𝐸𝑒

5Φ/2Lmatt,

(A.2)

where the ten-dimensional scalar curvature ̂R transformsaccording to the conformal transformation as

̂R = 𝑒−Φ/2

(R𝐸−

9

2

𝐺

𝐸𝑀𝑁∇𝑀∇𝑁Φ

−

9

2

𝐺


𝑁Φ) .

(A.3)

By substituting expression (A.3) in (A.2) we obtain

𝑆 =

1

2𝜅2

10

∫𝑑

10𝑋√−𝐺

𝐸(R

𝐸−

9

2

𝐺

𝐸𝑀𝑁∇𝑀∇𝑁Φ

−

1

2

𝐺


𝑁Φ) + ∫𝑑

10𝑋√−𝐺

𝐸𝑒

5Φ/2Lmatt.

(A.4)

The last expression is the ten-dimensional action in theEinstein frame. Expressing the metric determinant in four-dimensions in terms of the moduli field 𝜎 we have

det̂𝐺𝑀𝑁

=̂𝐺 = 𝑒

−12𝜎�̂�. (A.5)

By substituting the last expression (A.5) in (1) andconsidering that

̂R(10)

=̂R(4)

− 42�̂�

𝜇]∇𝜇𝜎∇]𝜎 + 12�̂�

𝜇]∇𝜇∇]𝜎, (A.6)

we obtain that

𝑆 =

1

2𝜅2

10

∫𝑑

4𝑥𝑑

6𝑦√−�̂�𝑒

−6𝜎𝑒

−2Φ[̂R

(4)

− 42�̂�

𝜇]∇𝜇𝜎∇]𝜎 + 12�̂�

𝜇]∇𝜇∇]𝜎 + 4�̂�

𝜇]∇𝜇Φ∇]Φ] .

(A.7)

Let us redefine in the last expression the dilaton field Φ as

Φ = 𝜙 −

1

2

ln (̂𝑉) , (A.8)

where ̂𝑉 = ∫𝑑6𝑦. So, expression (A.7) can be written as

𝑆 =

1

2𝜅2

10

∫𝑑

4𝑥√−�̂�𝑒

−2(𝜙+3𝜎)[̂R

(4)

− 42�̂�

𝜇]∇𝜇𝜎∇]𝜎

+ 12�̂�

𝜇]∇𝜇∇]𝜎 + 4�̂�

𝜇]∇𝜇𝜙∇]𝜙] .

(A.9)


The four-dimensional metric �̂�𝜇] means that the metric

is in the string frame, and we should take a conformaltransformation linking the String and Einstein frames. Welabel by 𝑔

𝜇] the metric of the external space in the Einsteinframe; this conformal transformation is given by

�̂�𝜇] = 𝑒

2Θ𝑔𝜇], (A.10)

which after some algebra gives the four-dimensional scalarcurvature:

̂R(4)

= 𝑒

−2Θ(R

(4)− 6𝑔

𝜇]∇𝜇∇]Θ − 6𝑔

𝜇]∇𝜇Θ∇]Θ) , (A.11)

where the function Θ is given by the transformation:

Θ = 𝜙 + 3𝜎 + ln(𝜅

2

10

𝜅2

4

) . (A.12)

Now, wemust replace expressions (A.10), (A.11), and (A.12) inexpression (A.9) and we find

𝑆 =

1

2𝜅2

4

∫𝑑

4𝑥√−𝑔 (R − 6𝑔

𝜇]∇𝜇∇]𝜙 − 6𝑔

𝜇]∇𝜇∇]𝜎

− 2𝑔

𝜇]∇𝜇𝜙∇]𝜙 − 96𝑔

𝜇]∇𝜇𝜎∇]𝜎 − 36𝑔

𝜇]∇𝜇𝜙∇]𝜎) .

(A.13)

At first glance, it is important to clarify one point related tothe stress-energy tensor which has the matrix form (6); thistensor belongs to the String frame. In order towrite the stress-energy tensor in the Einstein frame we need to work with thelast integrand of expression (1). So, after taking the variationwith respect to the ten-dimensional metric and opening theexpression we see that

∫𝑑

10𝑋√−̂𝐺𝑒

−2Φ𝜅

2

10𝑒

2Φ̂𝑇𝑀𝑁̂𝐺

𝑀𝑁

= ∫𝑑

4𝑥√−𝑔𝑒

2(Θ+𝜙)̂𝑇𝑀𝑁̂𝐺

𝑀𝑁

= ∫𝑑

4𝑥√−𝑔 (𝑒

2𝜙̂𝑇𝜇]𝑔

𝜇]+ 𝑒

2(Θ+𝜙)̂𝑇𝑚𝑛�̂�

𝑚𝑛) ,

(A.14)

where we can observe that the four-dimensional stress-energy tensor in the Einstein frame is defined as we said inexpression (7).

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was partially supported by CONACYT 167335,179881 Grants and PROMEPGrants UGTO-CA-3 andDAIP-UG640/2015.Thiswork is part of the collaborationwithin theInstituto Avanzado de Cosmologı́a and Red PROMEP: Grav-itation and Mathematical Physics under project QuantumAspects of Gravity in Cosmological Models, Phenomenologyand Geometry of Spacetime. One of authors (L. ToledoSesma) was supported by a Ph.D. scholarship in the graduateprogram by CONACYT.

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