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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
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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
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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
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4 Advances in High Energy Physics
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.
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Advances in High Energy Physics 5
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)
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6 Advances in High Energy Physics
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)
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Advances in High Energy Physics 7
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)
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8 Advances in High Energy Physics
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)
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Advances in High Energy Physics 9
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
𝑖𝑎2𝛽++ 𝜂
2𝑒
−𝑖𝑎2𝛽+]
⋅ [𝑐1𝑒
𝑖𝑎3𝛽−+ 𝑐
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
𝑖𝑎2𝛽++ 𝜂
2𝑒
−𝑖𝑎2𝛽+] [𝑐
1𝑒
𝑖𝑎3𝛽−+ 𝑐
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
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10 Advances in High Energy Physics
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
𝐸𝑀𝑁∇𝑀Φ∇
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
𝐸𝑀𝑁∇𝑀Φ∇
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)
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Advances in High Energy Physics 11
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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