Time-dependent toroidal compactification proposals and the Bianchi type I model: classical and quantum solutions

0.24 MB·English

by  L. Toledo Sesma

Time-dependent toroidal compactification proposals and the Bianchi type I model:
classical and quantum solutions
L. Toledo Sesma, J. Socorro, and O. Loaiza
∗ † ‡
Departamento de F´ısica,
DCI. Universidad de Guanajuato. Campus Le´on.
Loma del Bosque No. 103 Col. Lomas del Campestre C.P 37150 A.Postal E-143
Le´on, Guanajuato, Mexico.
(Dated: May 18, 2016)
6
Abstract
1
0
In this work we construct an effective four-dimensional model by compactifying a ten-dimensional
2
theoryof gravity coupled with areal scalar dilaton field on atime-dependenttorus. This approach

y isappliedtoanisotropiccosmologicalBianchitypeImodelforwhichwestudytheclassicalcoupling
a of the anisotropic scale factors with the two real scalar moduli produced by the compactification
M
process. Underthisapproach,wepresentanisotropizationmechanismfortheBianchiIcosmological
model through the analysis of the ratio between the anisotropic parameters and the volume of the

7 Universe which in general keeps constant or runs into zero for late times. We also find that the
1 presenceofextradimensionsinthismodelcanacceleratetheisotropizationprocessdependingonthe
momenta moduli values. Finally, we present some solutions to the corresponding Wheeler-DeWitt
]
(WDW)equation in thecontext of Standard Quantum Cosmology.
c
q
- PACSnumbers: 98.80.Qc,04.50.-h,04.20.Jb,04.50.Gh
r Keywords: Exactsolutions,classicalandquantum cosmology,dimensionalreduction
g
[

3
v
9 I. INTRODUCTION
7
7
The 2015releaseofPlanckdata hasprovideda detailedmapofcosmicmicrowavebackground(CMB) temperature
2
and polarization allowing us to detect deviations from an isotropic early Universe [1]. The evidence given by these
0
dataleadsustothepossibilitytoconsiderthatthereisnoexactisotropy,sincethereexistsmallanisotropydeviations
.
1 of the CMB radiation and apparent large angle anomalies. In that context, there have been recent attempts to fix
0
constraints on such deviations by using the Bianchi anisotropic models [2]. The basic idea behind these models is to
5
consider the present observational anisotropies and anomalies as imprints of an early anisotropic phase on the CMB
1
: which in turn can be explained by the use of different Bianchi models. In particular, Bianchi I model seems to be
v
related to large angle anomalies [2] (and references therein).
i
X
In a different context, attempts to understand diverse aspects of cosmology, as the presence of stable vacua
r
a and inflationary conditions, in the framework of supergravity and string theory have been considered in the last
years [3–10]. One of the most interesting features emerging from these type of models consists on the study of the
consequences of higher dimensional degrees of freedom on the cosmology derived from four-dimensional effective
theories [11, 12]. The usual procedure for that is to consider compactification on generalized manifolds, on which
internal fluxes have back-reacted,altering the smooth Calabi-Yau geometry and stabilizing all moduli [10].
The goal of the present work is to consider in a simple model some of the above two perspectives, i.e., we shall
considerthepresenceofextradimensionsinaBianchiImodelwiththepurposetotrackdowntheinfluenceofmoduli
fields in its isotropization. For that we shall consider an alternative procedure concerning the role played by moduli.
In particular we shall not consider the presence of fluxes, as in string theory, in order to obtain a moduli-dependent
scalar potential in the effective theory. Rather, we are going to promote some of the moduli to time-dependent fields
∗Electronicaddress: ltoledo@fisica.ugto.mx
†Electronicaddress: socorro@fisica.ugto.mx
‡Electronicaddress: oloaiza@fisica.ugto.mx
2
by considering the particular case of a ten-dimensional gravity coupled to a time-dependent dilaton compactified on
a 6-dimensionaltorus witha time-dependent K¨ahlermodulus. With the purposeto study the influence ofsuchfields,
we are going to ignore the dynamics of the complex structure modulos (for instance, by assuming that it is already
stabilizedbythe presenceofastring fieldinhigherscales). This shallallowsus toconstructclassicaleffective models
with two moduli.
However, we are also interested in possible quantum aspects of our model. Quantum implications on cosmology
from more fundamental theories are expected due to different observations. For instance, it has been pointed out
that the presence of extra dimensions leads to an interesting connection to the ekpyrotic model [13], which generated
considerable activity [8, 9, 14]. The essential ingredient in these models (see for instance [11]) is to consider an
effective action with a graviton and a massless scalar field, the dilaton, describing the evolution of the Universe,
while incorporating some of the ideas of pre-big-bang proposal [15, 16] in that the evolution of the Universe began
in the far past. On the other hand, it is well known that relativistic theories of gravity such as general relativity
or string theories are invariant under reparametrization of time. Quantization of such theories presents a number
of problems of principle known as “the problem of time” [17, 18]. This problem is present in all systems whose
classical version is invariant under time reparametrization, leading to its absence at the quantum level. Therefore,
the formalquestioninvolveshowto handle the classicalHamiltonianconstraint, 0,inthe quantumtheory. Also,
H≈
connectedwiththeproblemoftimeisthe“Hilbertspaceproblem”[17,18]referringtothenotatallobviousselection
of the inner product of states in quantum gravity, and whether there is a need for such a structure at all. The
abovefeatures,asitiswellknown,pointoutthenecessitytoconstructaconsistenttheoryofgravityatquantumlevel.
Analysis of effective four-dimensional cosmologies derived from M-theory and Type IIA string theory were
considered in [19–21] where string fluxes are related to the dynamical behaviour of the solutions. A quantum
description of the model was studied in [22, 23] where a flat Friedmann-Robertson-Walker geometry was considered
for the extended four-dimensional space-time while a geometry given by S1 T6 was assumed for the internal
×
seven-dimensional space, however the dynamical field are in the quintom scheme, since one of the fields have a
negative energy. Under a similar perspective, we study the Hilbert space of quantum states on a Bianchi I geometry
withtwotime-dependentscalarmoduliderivedfromaten-dimensionaleffectiveactioncontainingthedilatonandthe
Ka¨hler parameter from a six-dimensional torus compactification. We find that in this case, the wave function of this
Universe is represented by two factors, one depending on moduli while the second one depending on gravitational
fields. This behaviour is a property of all cosmologicalBianchi Class A models.
The work is organized in the following form. In section II we present the construction of our effective action by
compactificationonatime-dependenttorus,whileinSectionIIIwestudyitsLagrangianandHamiltoniandescriptions
usingastoymodeltheBianchitypeIcosmologicalmodel. SectionIVisdevotedonfindingthecorrespondingclassical
solutions for few different cases involving the presence or absence of matter content and the cosmological constant
term. Usingtheclassicalsolutionsfoundinprevioussections,wepresentanisotropizationmechanismforthe Bianchi
I cosmological model, through the analysis of the ratio between the anisotropic parameters and the volume of the
Universe, showing that in all the cases we have studied, its value keeps constant or runs into zero for late times.
In Section V we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of
Standard Quantum Cosmology and finally our conclusions are presented in Section VI.
II. EFFECTIVE MODEL
We start from a ten-dimensional action coupled with a dilaton (which is the bosonic component common to all
superstring theories), which after dimensional reduction can be interpreted as a Brans-Dicke like theory [24]. In
the string frame, the effective action depends on two spacetime-dependent scalar fields: the dilaton Φ(xµ) and the
Ka¨hler modulus σ(xµ). For simplicity, in this work we shall assume that these fields only depend on time. The
high-dimensional (effective) theory is therefore given by
1
S = d10X Gˆe 2Φ ˆ(10)+4GˆMN Φ Φ + d10X Gˆ ˆ (1)
2κ210 Z q− − "R ∇M ∇N # Z q− Lm
where all quantities qˆrefer to the string frame while the ten-dimensional metric is described by
ds2 =Gˆ dXMdXN =gˆ dxµdxν +h dymdyn, (2)
MN µν mn
3
where M,N,P,... are the indices of the ten-dimensional space, greek indices µ,ν,... = 0,...,3 and latin indices
m,n,p,... = 4,...,9 correspond to the external and internal space, respectively. We will assume that the six-
dimensional internal space has the form of a torus with a metric given by
h =e 2σ(t)δ , (3)
mn − mn
with σ a real parameter.
Dimensionally reducing the first term in (1) to four-dimensions in the Einstein frame (see Appendix A for details),
gives
1
S = d4x√ g 2gµν φ φ 96gµν σ σ 36gµν φ σ , (4)
4 2κ2 − R− ∇µ ∇ν − ∇µ ∇ν − ∇µ ∇ν
4 Z (cid:18) (cid:19)
where φ=Φ+ 1ln(Vˆ), with Vˆ given by
2
Vˆ =e6σ(t)Vol(X )= d6y. (5)
6
Z
By considering only a time-dependence on the moduli, one can notice that for the internal volume Vol(X ) to be
6
small, the modulus σ(t) should be a monotonic increasing function on time (recall that σ is a real parameter), while
Vˆ is time and moduli independent.
Now, concerning the second term in (1), we shall requiere to properly define the ten-dimensional stress-energy
tensor Tˆ . In the string frame it takes the form
MN
Tˆ 0
Tˆ = µν , (6)
MN 0 Tˆ
(cid:18) mn(cid:19)
where Tˆ and Tˆ denote the four and six-dimensional components of Tˆ . Observe that we are not considering
µν mn MN
mixing components of Tˆ among internal and external components although non-constrainedexpressions for Tˆ
MN MN
have been considered in [25]. In the Einstein frame the four-dimensional components are given by
T =e2φTˆ . (7)
µν µν
It is important to remark that there are some dilemmas about what the best frame to describe the gravitational
theory is. Here in this work we have taken the Einstein frame. Useful references to find a discussion about String
and Einstein frames and its relationship in the cosmologicalcontext are [26–30].
Now with the expression(4)we proceedto build the Lagrangianandthe Hamiltonian ofthe theoryat the classical
regime employing the anisotropiccosmologicalBianchi type I model. The moduli fields will satisfy the Klein-Gordon
like equation in the Einstein frame as an effective theory.
III. CLASSICAL HAMILTONIAN
In order to construct the classicalHamiltonian, we are going to assume that the backgroundof the extended space
isdescribedbyacosmologicalBianchitypeImodel. Forthat,letusrecallherethecanonicalformulationintheADM
formalism of the diagonal Bianchi Class A cosmologicalmodels; the metric has the form
ds2 = N(t)dt2+e2Ω(t)(e2β(t)) ωiωj, (8)
ij
−
where N(t) is the lapse function, β (t) is a 3 3 diagonal matrix, β = diag(β +√3β ,β √3β , 2β ), Ω(t)
ij ij + + +
and β are scalarfunctions, knownas Misner v×ariables,ωi are one-formsthat characteriz−eeach−cosmo−log−icalBianchi
type m±odel[31], andobey the formdωi = 1Ci ωj ωk, withCi the structureconstantsofthe correspondingmodel.
2 jk ∧ jk
The one-forms for the Bianchi type I model are ω1 =dx,ω2 =dy, and ω3 =dz. So, the corresponding metric of the
Bianchi type I in Misner’s parametrization has the form
ds2 = N2dt2+R2dx2+R2dy2+R2dz2, (9)
I − 1 2 3
4
where R 3 are the anisotropic radii and they are given by
{ i}i=1
R1 =eΩ+β++√3β−, R2 =eΩ+β+−√3β−, R3 =eΩ−2β+, V =R1R2R3 =e3Ω,
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 a
structure , corresponding to an energy-momentum tensor of perfect fluid [32–35]
T =(p+ρ)u u +pg
µν µ ν µν
that satisfies the conservation law Tµν = 0. Taking the equation of state p = γρ between the energy density and
ν
∇
the pressure of the comovil fluid, a solution is given by ρ = C e 3(1+γ)Ω with C the corresponding constant for
γ − γ
different values of γ related to the Universe evolution stage. Then, the Lagrangiandensity reads
=16πG √ gρ+2√ gΛ=16NπG C e 3(1+γ)Ω+2NΛ,e3Ω
mat N N γ −
L − −
while the Lagrangianthat describes the fields dynamics is given by
e3Ω
= 6Ω˙2 6β˙2 6β˙2 +96σ˙2+36φ˙σ˙ +2φ˙2+16πGN2ρ+2ΛN2 , (10)
LI N − +− −
h i
using the standard definition of the momenta, Πqµ = ∂∂qL˙µ, where qµ =(Ω,β+,β−,φ,σ), we obtain
12e3ΩΩ˙ Ne 3ΩΠ
Π = , Ω˙ = − Ω,
Ω
N 12
12e3Ωβ˙ Ne 3ΩΠ
Π = ±, β˙ = − β±,
± − N ± − 12
e3Ω Ne 3Ω
Π = 36σ˙ +4φ˙ , φ˙ = − [3Π 16Π ],
φ σ φ
N 44 −
e3Ω h i Ne 3Ω
Π = 192σ˙ +36φ˙ , σ˙ = − [9Π Π ],
σ φ σ
N 132 −
and, introducing them into the Lagrangianhdensity, weiobtainthe canonicalLagrangianas =Π q˙µ N .
canonical qµ
L − H
WhenweperformthevariationofthiscanonicallagrangianwithrespecttoN, δLcaδnNonical =0,implyingtheconstraint
= 0. In our model the only constraint corresponds to Hamiltonian density, which is weakly zero. So, we obtain
I
H
the Hamiltonian density for this model
e 3Ω 48 18 1
= − Π2 Π2 Π2 Π2 + Π Π Π2 384πG ρ e3(1 γ)Ω 48Λe6Ω . (11)
HI 24 Ω− +− −− 11 φ 11 φ σ− 11 σ− N γ − −
(cid:20) (cid:21)
We introduce a new setof variablesin the gravitationalpartgivenby eβ1+β2+β3 =e3Ω =V whichcorrespondsto the
volume of the Bianchi type I Universe, in similar way that the flat Friedmann-Robetson-Walker metric (FRW) with
a scale factor. This new set of variables depends on Misner variables as
β = Ω+β +√3β ,
1 +
−
β = Ω+β √3β , (12)
2 +
− −
β = Ω 2β ,
3 +
−
from which the Lagrangiandensity (10) can be transformed as
eβ1+β2+β3
I = 2φ˙2+36φ˙σ˙ +96σ˙2+2β˙1β˙2+2β˙1β˙3+2β˙3β˙2+2ΛN2+16πGN2ργe−(1+γ)(β1+β2+β3) , (13)
L N
(cid:18) (cid:19)
and the Hamiltonian density reads
1 16 6 1
HI = 8e−(β1+β2+β3) −Π21−Π22−Π23+2Π1Π2+2Π1Π3+2Π2Π3− 11Π2φ+ 11ΠφΠσ− 33Π2σ
(cid:20)
128πGργe(1−γ)(β1+β2+β3) 16Λe2(β1+β2+β3) . (14)
− −
(cid:21)
So far, we have built the classical Hamiltonian density from a higher dimensional theory; this Hamiltonian contains
a barotropic perfect fluid, we have explicitly added. The next step is to analyze three different cases involving terms
in the classical Hamiltonian (14), and find solutions to each of them in the classical regime.
5
A. Isotropization
The current observations of the cosmic background radiation set a very stringent limit to the anisotropy of the
Universe [37], therefore it is important to consider the anisotropy of the solutions. We shall denote through all our
study derivatives of functions F with respect to τ = Nt by F (the following analysis is similar to this presented
′
recently in the K-essence theory, since the corresponding action is similar to this approach [36]). Recalling the
Friedmann like equation to the Hamiltonian density
1 8 Λ
Ω2 =β2+β2+16σ2+ φ2+6φσ + πGµ e 3(1+γ)Ω+ , (15)
′ +′ −′ ′ 3 ′ ′ ′ 3 γ − 3
we can see that isotropization is achieved when the terms with β2 go to zero or are negligible with respect to
′
the other terms in the differential equation. We find in the literatu±re the criteria for isotropization, among others,
(β2+β2)/H2 0, (β2+β2)/ρ 0 , that are consistent with our above remark. In the present case the
com+′ pari−s′on with t→he density+′ shou−′ld includ→e the contribution of the scalar field. We define an anisotropic density ρ ,
a
that is proportional to the shear scalar,
ρ =β2+β2, (16)
a +′ ′
−
and will compare it with ρ , ρ , and Ω2. From the Hamilton analysis we now see that
γ φ ′
Π˙ = 0, Π =p =constant,
φ φ φ
→
Π˙ = 0, Π =p =constant,
σ σ σ
→
Π˙ = 0, Π =p =constant,
+ + +
→
Π˙ = 0, Π =p =constant,
− → − −
and the kinetic energy for the scalars fields φ and σ are proportional to e 6Ω. This can be seen from definitions
−
of momenta associated to the Lagrangian (10) and the above equations. Then, defining κ as κ2 p2 +p2 +
16 (p 9p )2+ 1 (3p 16p )2+ 1 (p 9p )(3p 16p ), we have that Ω Ω ∼ + −
1322 σ− φ 3442 σ− φ 2244 σ − φ σ− φ
· ·
Λ 8
ρa e−6Ω, ρφ,σ e−6Ω, Ω′2 +κΩ2e−6Ω+bγe−3(1+γ)Ω, bγ = πGNµγ. (17)
∼ ∼ ∼ 3 3
With the use of these parameters, we find that the following ratios :
ρ ρ ρ 1
ρφa,σ ∼constant, ργa ∼e3Ω(γ−1), Ω′a2 ∼ κΩ2+ Λ3e6Ω+bγe3(1−γ)Ω. (18)
We observe that for an expanding Universe the anisotropic density is dominated by the fluid density (with the
exception of the stiff fluid) or by the Ω2 term and then at late times the isotropization is obtained since the above
′
ratios tends to zero.
IV. CASE OF INTEREST
In this section we present the classical solutions for the Hamiltonian density of the Lagrangian (10) we have
previously built in terms of a new set of variables (12), focusing on three different cases. We start our analysis on
the vacuum case, and on the case with a cosmological term Λ. Finally we shall analyze the general case considering
matter content and a cosmologicalterm.
A. Vacuum case
To analyze the vacuum case, we will take in the Hamiltonian density (14) that ρ =0 and Λ=0, obtaining that
γ
1 16 6 1
HIvac = 8e−(β1+β2+β3) −Π21−Π22−Π23+2Π1Π2+2Π1Π3+2Π2Π3− 11Π2φ+ 11ΠφΠσ − 33Π2σ . (19)
(cid:20) (cid:21)
6
The Hamilton equations for the coordinates fields q˙ = ∂ and the corresponding momenta P˙ = ∂ becomes
i ∂PHi i ∂Hqi
Π˙ = 0, Π =constant,
i i
−H≡ ⇒
Π˙ = 0, Π =constant,
φ φ
⇒
Π˙ = 0, Π =constant,
σ σ
⇒
1
β1′ = 4e−(β1+β2+β3) [−Π1+Π2+Π3],
1
β2′ = 4e−(β1+β2+β3) [−Π2+Π1+Π3], (20)
1
β3′ = 4e−(β1+β2+β3) [−Π3+Π1+Π2],
φ′ = φ0e−(β1+β2+β3),
σ′ = σ0e−(β1+β2+β3).
The gravitational momenta are constant by mean the Hamiltonian constraint, first line in the last equation. In this
way, the solution for the sum of β functions become
i
β +β +β =ln[ǫτ +b ], ǫ=b +b +b ,
1 2 3 0 1 2 3
with b =Π and b an integration constant. Therefore, Misner variables are expressed as
i i 0
Ω=ln(ǫτ +b0)13 , β+ =ln(ǫτ +b0)ǫ−63ǫb3 , β =ln(ǫτ +b0)b3+2√2b31ǫ−ǫ , (21)
−
while moduli fields are given by
φ σ
0 0
φ= ln(ǫτ +b ), σ = ln(ǫτ +b ),
0 0
ǫ ǫ
where φ = Π and σ = Π . Notice that in this case, the associated external volume given by e3Ω and the moduli
0 φ 0 σ
fields φ and σ are all logarithmically increasing functions on time, implying for the later that the internal volume
shrinks in size for lately times, as expected, while the four-dimensionalUniverse expands into an isotropic flat space-
time. The parameter ρ , goes to zero at very late times independently of b .
a 0
B. Cosmological term Λ
The corresponding Hamiltonian density becomes
1 16 6
HI = 8e−(β1+β2+β3) −Π21−Π22−Π23+2Π1Π2+2Π1Π3+2Π2Π3− 11Π2φ+ 11ΠφΠσ
(cid:20)
1
Π2 16Λe2(β1+β2+β3) , (22)
−33 σ−
(cid:21)
and the corresponding Hamilton’s equations are
Π′1 =Π′2 =Π′3 = 4Λeβ1+β2+β3,
1
β1′ = 4e−(β1+β2+β3) [−Π1+Π2+Π3],
1
β2′ = 4e−(β1+β2+β3) [−Π2+Π1+Π3], (23)
1
β3′ = 4e−(β1+β2+β3) [−Π3+Π1+Π2],
φ′ = φ0e−(β1+β2+β3),
σ′ = σ0e−(β1+β2+β3),
where the constants φ and σ are the same as in previous case. To solve the last system of equations we shall take
0 0
the following ansatz
Π =Π +δ =Π +δ , (24)
1 2 2 3 3
7
with δ and δ constants. By substituing into the Hamiltonian (22) we find a differential equation for the momenta
2 3
Π given by
1
1
ΛΠ′12−3Π21+νΠ1+δ1 =0, (25)
where the corresponding constants are
ν =2(δ +δ ), δ (q )=(δ δ )2 q , (26)
2 3 1 0 2 3 0
− −
with the constant (related to the momenta scalar field)
16 6 1
q = Π2 Π Π + Π2. (27)
0 11 φ− 11 φ σ 33 σ
The solution for Π is thus
1
ν 1
Π = + ν2+12δ cosh √3Λτ . (28)
1 1
6 6
p (cid:16) (cid:17)
Hence, using the last result (28) in the rest of equations in (23) we obtain that
1 ν2+12δ
β +β +β =ln 1 sinh(√3Λτ) , (29)
1 2 3
8 3Λ
r !
and the corresponding solutions to the moduli fields are
8φ √3Λ
2
φ = ln tanh τ , (30)
√ν2+12δ 2
1 (cid:18) (cid:18) (cid:19)(cid:19)
8σ √3Λ
2
σ = ln tanh τ .
√ν2+12δ 2
1 (cid:18) (cid:18) (cid:19)(cid:19)
With the last results we see that the solutions to the Misner variables (Ω,β ) are given by
±
1 1 ν2+12δ
Ω = ln 1 sinh(√3Λτ) ,
3 8 3Λ
r !
2 (δ 2δ ) √3Λ
2 3
β = − ln tanh τ , (31)
+
3 √ν2+12δ1 2 !
2 δ √3Λ
2
β = ln tanh τ .
− −√3 √ν2+12δ1 2 !
and the isotropization parameter ρ is given by
a
Λ (δ 2δ )2+δ2 √3Λ
ρ = 2− 3 2 coth2 τ . (32)
a 3 ν2+12((δ δ )2 q ) 2
2 3 0 !
− −
Notice that the external four-dimensional volume expands as time τ runs and it is affected wether q is positive or
0
negative. This in turns is determined by how faster evolves the moduli fields with respect to each other (i.e., wether
Π is largerorsmaller than Π ). Therefore,isotropizationis reachedindependently of the values ofq but expansion
φ σ 0
of the Universe (and shrinking of the internal volume) is affected by it.
C. Matter content and cosmological term
For this case, the correspondingHamiltonian density becomes the equation(14). So, using the Hamilton equation,
wecanseethatthe momentaassociatedtothe scalarsfields φ andσ areconstants,andweshalllabelthese constants
8
by φ and σ , respectively.
1 1
Π′1 =Π′2 =Π′3 = 4Λeβ1+β2+β3 +16πG(1−γ)ργe−γ(β1+β2+β3),
1
β1′ = 4e−(β1+β2+β3) [−Π1+Π2+Π3],
1
β2′ = 4e−(β1+β2+β3) [−Π2+Π1+Π3], (33)
1
β3′ = 4e−(β1+β2+β3) [−Π3+Π1+Π2],
φ′ = φ2e−(β1+β2+β3),
σ′ = σ2e−(β1+β2+β3).
Follow the same steps as the section IVB, we see that is possible to relate the momenta associated to β , β , and β
1 2 3
in the following way
Π =Π +k =Π +k (34)
1 2 1 3 2
also, the differential equation for fields φ and σ can be reduced to quadratures as
φ=φ2 e−(β1+β2+β3)dτ, σ =σ2 e−(β1+β2+β3)dτ. (35)
Z Z
We proceed to consider specific values for γ.
1. γ =±1 cases
Let us introduce the generic parameter
4Λ, γ =1
λ= (36)
4Λ+32πGρ , γ = 1
1
(cid:26) − −
for which Π′1 = λeβ1+β2+β3. Substituting this into the Hamiltonian (14) we find the differential equation for the
momenta Π as
1
4
λΠ′12−3Π21+νΠ1+δ1 =0, (37)
with the solution given by Π is
1
ν 1 √3λ
Π = + ν2+12δ cosh τ . (38)
1 1
6 6 2
!
p
On the other hand, using the last result (38) we obtain from Hamilton equations that
1 δ +12δ √3λ
3 1
β +β +β = ln sinh τ . (39)
1 2 3
4 3λ 2
r !!
The corresponding solutions to the moduli fields (35) can be found as
8φ √3λ 8σ √3λ
2 2
φ= ln tanh τ , σ = ln tanh τ . (40)
√ν2+12δ1 4 ! √ν2+12δ1 4 !
The solutions to the Misner variables (12), (Ω,β ) are given by
±
1 1 δ2+12δ √3λ
1
Ω = ln sinh τ ,
3 4 3λ 2
r !!
2 (δ 2δ ) √3λ 1 1 ν2+12δ
2 3 1
β = − ln tanh τ + ln , (41)
+
3 √ν2+12δ1 2 ! 6 4r 3λ !
2 δ √3λ 1 1 ν2+12δ
2 1
β = ln tanh τ ln .
− −√3 √ν2+12δ1 2 !− 2√3 4r 3λ !
9
It is remarkable that equations (35) for the fields φ and σ are maintained for all Bianchi Class A models, and in
particular, when we use the gauge N = eβ1+β2+β3, the solutions for these fields are independent of the cosmological
models, whose solutions are φ= φ0t and σ = σ0t.
8 8
After studying the γ =0 case we shall comment on the physical significance of these solutions.
2. γ =0 case
The Hamilton equations are
Π′1 =Π′2 =Π′3 = 4Λeβ1+β2+β3 +α0, α0 =16πGρ0
Π = 0, Π =φ =cte,
′φ φ 1
(42)
from which the momenta Π fulfils the equation
1
1
ΛΠ′12−3Π21+µΠ1+ξ1 =0, (43)
where
α2
ξ =δ 0. (44)
1 1
− Λ
The solution is given by
ν 1
Π = + ν2+12ξ cosh √3Λτ . (45)
1 1
6 6
p (cid:16) (cid:17)
Using this result in the corresponding Hamilton equations, we obtain that
1 ν2+12ξ
β +β +β =ln 1 sinh √3Λτ , (46)
1 2 3
8 3Λ
r !
(cid:16) (cid:17)
and the corresponding solutions to the dilaton field φ and the moduli field σ are given by
8φ √3Λ 8σ √3Λ
2 2
φ= lntanh τ , σ = lntanh τ . (47)
ν2+12ξ1 2 ! ν2+12ξ1 2 !
p p
So, the solutions to the Misner variables (Ω,β ), are
±
1 1 ν2+12ξ
Ω = ln 1 sinh √3Λτ ,
3 8 3Λ
r !
(cid:16) (cid:17)
2 δ 2δ √3Λ
2 3
β = − lntanh τ , (48)
+
3 ν2+12ξ1 2 !
p2 δ √3Λ
2
β = lntanh τ .
− −√3 ν2+12ξ1 2 !
p
For both solutions, γ =0, 1, we also observe that for positive (negative) q , the Universe expands slower (faster)
0
±
than in the absence of extra dimensions while the internal space shrinks faster (slower) for positive (negative) q . In
0
this case, isotropization parameters are also accordingly affected by q , through relations (27).
0
So far, our analysis has been completely classic. Now, in the next section we deal with the quantum scheme and
we are going to solve the WDW equation in standard quantum cosmology.
10
V. QUANTUM SCHEME
Solutions to the Wheeler-DeWitt (WDW) equation dealing with different problems have been extensively used
in the literature. For example, the important quest of finding a typical wave function for the Universe, was nicely
addressedin[38],whilein[39]thereappearsanexcellentsummaryconcerningtheproblemofhowaUniverseemerged
from a big bang singularity, cannot longer be neglected in the GUT epoch. On the other hand, the best candidates
for quantum solutions become those that have a damping behavior with respect to the scale factor, represented in
our modelwith the Ω parameter,inthe sense that we obtaina goodclassicalsolutionusing the WKB approximation
in any scenario in the evolution of our Universe [40]. The WDW equation for this model is achievedby replacing the
momentaΠ = i∂ , associatedtothe Misnervariables(Ω,β ,β )andthemodulifields (φ,σ) intheHamiltonian
qµ qµ +
(11). The factor−e 3Ω may be factor ordered with Πˆ in many way−s. Hartle and Hawking [40] have suggested what
− Ω
might be called a semi-general factor ordering which in this case would order e 3ΩΠˆ2 as
− Ω
−e−(3−Q)Ω∂Ωe−QΩ∂Ω =−e−3Ω∂Ω2 +Qe−3Ω∂Ω , (49)
whereQisanyrealconstantthatmeasuretheambiguityinthefactororderinginthevariableΩandthecorresponding
momenta. We will assume in the following this factor ordering for the Wheeler-DeWitt equation, which becomes
∂Ψ 48∂2Ψ 1 ∂2Ψ 18 ∂2Ψ
(cid:3)Ψ+Q∂Ω − 11 ∂φ2 − 11 ∂σ2 + 11∂φ∂σ − bγe3(1−γ)Ω+48Λe6Ω Ψ=0, (50)
h i
where (cid:3) is the three dimensional d’Lambertian in the ℓµ = (Ω,β ,β ) coordinates, with signature ( ++). On
+
the other hand, we could interpreted the WDW equation (50) as a ti−me-reparametrization invariance−of the wave
function Ψ. At a glance, we can see that the WDW equation is static, this canbe understood as the problem of time
in standard quantum cosmology. We can avoid this problem by measuring the physical time with respect a kind of
time variable anchored within the system, that means that we could understand the WDW equation as a correlation
between the physical time and a fictitious time [41, 42]. When we introduce the ansatz
Ψ=Φ(φ,σ)ψ(Ω,β ), (51)
±
in (50), we obtain the general set of differential equations (under the assumed factor ordering),
∂ψ µ2
(cid:3)ψ+Q bγe3(1−γ)Ω+48Λe6Ω ψ = 0, (52)
∂Ω − − 5
(cid:20) (cid:21)
48 ∂2Φ 1 ∂2Φ 18 ∂2Φ
+ +µ2Φ=0. (53)
11 ∂φ2 11 ∂σ2 − 11 ∂φ∂σ
where we choose the separation constant µ2 for convenience to reduce the second equation. The solution to the
5
hyperbolic partial differential equation (53) is given by
Φ(φ,σ)=C sin(C φ+C σ+C )+C cos(C φ+C σ+C ), (54)
1 3 4 5 2 3 4 5
where C 5 are integration constants and they are in terms of µ. We claim that this solutions is the same for all
{ i}i=1
Bianchi Class A cosmological models, because the Hamiltonian operator in (50) can be written in separated way as
Hˆ(Ω,β ,φ,σ)Ψ=Hˆ (Ω,β )Ψ+Hˆ (φ,σ)Ψ=0, where Hˆ y Hˆ represents the Hamiltonian to gravitationalsector
g m g m
andthe±moduli fields, respe±ctively. To solveequation(52), we now setψ(Ω,β )= (Ω) (β ) (β ), obtaining the
1 + 2
following set of ordinary differential equations ± A B B −
d2 d
dΩA2 −QdAΩ + bγe−3(γ−1)Ω+48Λe6Ω+σ2 A=0, (55)
d2 h i
dβB21 +a22B1 =0, ⇒ B1 =η1eia2β+ +η2e−ia2β+, (56)
+
d2
dβB22 +a23B2 =0, ⇒ B2 =c1eia3β− +c2e−ia3β−, (57)
−
where σ2 =a2+a2+a2, c and η are constants. We now focus on the Ω dependent part of the WDW equation. We
1 2 3 i i
solve this equation when Λ=0 and Λ=0.
6