An Open Inflationary Model for Dimensional Reduction and its Effects on the Observable Parameters of the Universe

0.1 MB·English

by  Koray Karaca

February 5, 2008 6:35 WSPC/INSTRUCTION FILE paper
International JournalofModernPhysicsA
(cid:13)c WorldScientificPublishingCompany
5
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0 AN OPEN INFLATIONARY MODEL FOR DIMENSIONAL
2 REDUCTION AND ITS EFFECTS ON THE OBSERVABLE
n PARAMETERS OF THE UNIVERSE
a
J

1
KORAYKARACAandSELC¸UKBAYIN
1
Department of Physics, Middle East Technical University,06531, Ankara, Turkey

1 [e-mail], [e-mail]
v
0
0 Assuming that higher dimensions existed in the early stages of the universe where the
2 evolutionwasinflationary,weconstructanopen,singularity-free,spatiallyhomogeneous
1 andisotropiccosmologicalmodeltostudytheeffectsofdimensionalreductionthatmay
0 havetakenplaceduringtheearlystagesoftheuniverse.Weconsiderdimensionalreduc-
5 tiontotakeplaceinastepwisemannerandinterpreteachstepasaphasetransition.By
0 imposingsuitableboundaryconditions wetracetheireffects onthepresentdayparam-
/ etersoftheuniverse.
h
p Keywords:Dimensionalreduction;inflation;phasetransition.
-
o
r
t 1. Introduction
s
a During the past two decades, several closed cosmological models aiming the
:
v singularity-freedescriptionofthe universehavebeenproposed.1,2,3 Intheseworks,
i
X theuniversewasmodeledasaclosedFriedmann-Lemaitre-Robertson-Walkerspace-
r time, bouncing from a Planck mass and radius state with inflation, connecting to
a
thestandardmodelofradiationandthentotheeraofmatterdominance.Although
they correctly described the evolution of the universe, these models could not pre-
dicted the present value of the Hubble parameter within the observed range. The
3
approach developed by Bayın, Cooperstock and Faraoni was applied to open ge-
4
ometrybyKaracaandBayın anditwasshownthatasingularity-freecosmological
model which complies with the recentmeasurements is possible for open geometry.
Inthis model,since the initially staticcondition(a˙(0)=0)usedinthe closedmod-
els can no longer be used in the open universe case, the universe starts with an
initial expansion rate, i.e., a˙(0)6=0. Hence, this model is a two-parameteruniverse
modelinwhichoneoftheparametersdeterminesthestrengthoftheinitialvacuum
dominance and the other corresponds to the initial expansion rate.
It is well known that some theories require the existence of higher dimensions
in the early stagesof the universe.In this paper, we investigate the possibility that
higherdimensionsmighthaveexistedduringtheinflationaryprematterphaseofthe
universe and explore its observable consequences. As in the previous papers1,2,3,4
5 6
we adopt the Gliner -Markov picture of Planck limits to the physical quantities.
1
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2 K. Karaca and S. Bayın
2. Description of the Model
In our model, universe starts its journey as a D (D > 3) dimensional open Fried-
mann model (D denotes the number of initial space dimensions). At t = 0 density
of the universe is taken as the Planck density. Existence of higher dimensions and
theirreductiontothreedimensionsareassumedtotakeplaceduringtheinflationary
prematter phase where the equation of state is given as
P =(γ −1)ρ . (1)
D D D
Here,γ isassumedtobeapositive,smallanddimensionlessconstantthatcharac-
D
terizes the prematter in D dimensions, and P and ρ represent the pressure and
D D
energy density in D dimensions, respectively.
During this phase the universe expands isentropically while its temperature in-
creases.Thisexpansioncontinuesuntilthe maximumallowedtemperature,i.e.,the
Planck temperature of that dimension (T ) is reached. At this point, we pos-
pl,D
tulate that a phase transition to a lower dimension takes place, where the Planck
temperature is higher. Hence, the universe finds more room for further inflation.
ThisprocesscontinuesuntilwereachD =3.Atthispoint,eitherthelastreduction
maycarrytheuniversedirectlytothestandardradiationera(P = 1ρ)orprematter
3
era may continue once more in the usual three-dimensional space and then comes
the radiation era. In either case, the universe is eventually carried to the era of
matter dominance (P =0). In this work, we will consider the second alternative.
Naturally, neither the dimensional reduction takes place at an instant nor we
couldexpect the universeto remainhomogeneousandisotropicduring dimensional
reduction. However, we expect the duration of this process to be very short com-
pared to the times that the universe spends in constant D eras where it is ex-
panding as a D dimensionalFriedmann model. Details of what happens during the
dimensional reduction phase is beyond the scope of the present work. However, by
imposing suitable boundary conditions at the transition points we could trace the
effects of extra dimensions on the currently observable parameters of the universe,
i.e., Hubble constant, age and density.
3. Boundary Conditions and Solutions for the Scale Factor
Weassumeopen,homogeneousandisotropic(1+D)dimensionalspacetimeswhere
the metric is given as:
ds2 =c2dt2−a2 (t)[dχ2+sinh2χdθ2+sinh2χsin2θ dθ2+..... (2)
D 1 1 2
.....+sinh2χsin2θ1sin2θ2.....sin2θD−2dθD2−1],
where a (t) represents the scale factor of the universe, t is the comoving time,
D
0≤θn ≤π, 0≤θD−1 ≤2π,0≤χ≤∞ and n=1,2,.........,D−2. We also assume
thattheuniverseisfilledwithaperfectfluidwhichisrepresentedbyastress-energy
tensor having the following nonvanishing components:
T0 =ρ , T1 =T2 =.....=TD =−P . (3)
0 D 1 2 D D
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An OpenInflationary Model forDimensional Reduction 3
Einstein’s gravitational field equations in (1+D) dimensional space-time are (see,
e.g., Ref. 7, pp. 470-482)
2
D(D−1) a˙ A G
D D D
−1 =2 ρ , (4)
2a2 c c4 D
D "(cid:18) (cid:19) # D
2
a¨ (D−1)(D−2) a˙ A G
D D D D
(1−D) − −1 =2 P , (5)
c2 a 2a2 c c4 D
(cid:18) D D(cid:19) D "(cid:18) D(cid:19) # D
wherethesubscriptDdenotesthevalueofthatquantityinDdimensions,A isthe
D
solid angle subtended by a sphere in a D dimensional space (A = 4π, A = 2π2,
3 4
A = 8π2/3,...) and a dot denotes differentiation with respect to the cosmic time
5
t. We may combine Eqs. (1), (4) and (5) to obtain an equation involving only the
scale factor a(t). Furthermore, if we introduce a new dependent variable given by
u ≡ a′D = dlnaD and define conformal time as η = c t dt′ we get the Riccati
D aD dη a(t′)
equation
R
u′ +c u2 −c =0, (6)
D D D D
where c = Dγ −1 and a prime denotes differentiation with respect to η. In the
D 2 D
following, we will consider values of γ such that c 6= 0. Eq. (6) can easily be
D D
solved by setting u ≡ 1 wD′ = ln w1/cD ′ which gives the solution
D cDwD D
h (cid:16) (cid:17)i
a (η)=a sinh(c η+δ )1/cD, (7)
D 0,D D D
where we have introduced the subscript D to identify the era which it applies and
a and δ are the integration constants to be determined from the initial condi-
0,D D
tions.
We also assume that at the end of the prematter era the fluid pervading the
universe attained thermal equilibrium with a thermal spectrum. At this point we
takethedensityoftheuniverseequaltothatofamasslessscalarfieldwithathermal
spectrum at T , i.e.,
pl,D
A
ρ(η )= D Γ(D+1)ξ(D+1)(kT )D+1. (8)
D (2πc~)D pl,D
Then, it is possible to find the duration of the initial prematter era by making use
of Eqs. (4) and (8).
According to our consideration, dimensional reduction takes place via a first
orderphasetransition.But,its durationcouldbe takennegligiblysmallwhencom-
paredwiththedurationsoftheconstantDerasinthehistoryoftheuniverse.Thus,
dimensionalreductioncould be consideredto be taking place at aninstant of time.
Denoting this time as η , we may write Eq. (4) in terms of conformal time η just
D
before the dimensional reduction at η =η −|ǫ| as
D
D(D−1) a′ 2 1 2A G
D − = D Dρ , (9)
2 a2 a2 c4 D
"(cid:18) D(cid:19) D#
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4 K. Karaca and S. Bayın
where ǫ is a small number. After the dimensional reduction at η , the number of
D
space dimensions becomes (D−1) and Eq. (9) takes the following form
(D−1)2(D−2) aa2′D−1 2− a21 = 2AD−c14GD−1ρD−1, (10)
"(cid:18) D−1(cid:19) D−1#
at η = ηD+ | ǫ | . In Eqs. (9) and (10), GD and GD−1 denote the gravitational
constantinD andD−1dimensions,respectively.Atthispoint,asthemostnatural
′
and simple boundary conditions we take a (η) and a (η) as continuous physical
D D
quantitiesatthetransitionpoints.Ofcourse,strictlyspeakingneitherofthemcould
be continuous. However, since the phase transitions take place in a time interval
whose duration is expected to be significantly smaller than those of the constant
D eras and the evolution of the universe during these eras is inflationary, we could
argue that during the transition period the change in a (η) is negligible. On the
D
′
otherhand,inthelimitasǫ→0adiscontinuityina (η)wouldimplyinfiniteforces
D
acting on the universe, which in turn implies infinite values for the combinations
G ρ and G P during the dimensional reduction. Thus, for the time being we
D D D D
′
argue that taking a (η) as continuous at the transition points is a good working
D
assumption.We nowconsiderEqs.(9)and(10)in the limit asǫ→0andtaketheir
ratios to obtain
GD D AD−1 ρD−1
= . (11)
GD−1 D−2 AD (cid:20) ρD (cid:21)η=ηD
With the help of Eq. (11) we relate the change in G to that in ρ . Thus, in
D D
order to find how G changes as D changes, we have to know the discontinuity
D
in density when the universe experiences a dimensional reduction. We may express
this discontinuity by defining
ρD−1
lD,D−1 = , (12)
ρ
(cid:20) D (cid:21)η=ηD
whichgivesusa criticallengthparameterwith[lD,D−1]=cm.Now,wecouldwrite
the following expression for the change in G :
D
GD D AD−1
= lD,D−1, (13)
GD−1 D−2 AD
where lD,D−1 defines a characteristic length scale which marks the point at which
dimensional reduction occurs.
4. Numerical Results and Conclusion
Inordertodemonstratethebasicfeaturesofourmodelwenowconsideranumerical
modelwhichstartsfromaD =4premattereraandevolvesintoaD =3prematter.
In this model, to produce numerical results for the cosmological parameters of the
universe one has to assign numerical values to the characteristic parameters of the
model, i.e., c , c , l and the initial value of the Hubble constant H(0). It is
3 4 4,3
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An OpenInflationary Model forDimensional Reduction 5
apparent from Eqs. (1) and (7) that c and c represent the vacuum dominance of
3 4
the universe in D = 3 and D = 4 prematter eras, respectively. In determining the
critical parameters of the model we first consider the present value of the density:
2(2−c3)
ρ(η )=6.2269·10−35T 4l4,3 3(2c3−1) gr/cm3, (14)
now m
πl
(cid:18) pl (cid:19)
where T , whichstands for the recombinationtemperature,is roughlyatthe order
m
of103K (see,e.g.,Ref.7,pp.70-82).Furthermore,sincetheevolutionoftheuniverse
is inflationary during the prematter eras, we have
c and c ∈(−1,0). (15)
4 3
It is evident from Eqs. (14) and (15) that as the numerical value of the term
(4l /πl )getsfarfromunity,thepresentvalueofthedensitypredictedinEq.(14)
4,3 pl
falls rapidly below 10−31gr/cm3 which represents the order of the present value of
the totaldensity.Hence,we may conclude thata phase transitionis signalledwhen
the critical length parameter approachesthe Planck length of the lower dimension.
We also note that while our model is insensitive to the choice of c , its pre-
3
dictions strongly depend on the choice of c and H(0). We get numerical results
4
8
that comply with recent observations only when c gets numerical values close
4
to -1 and H(0) remains at the order of 1063 km/sec.mpc. As a specific exam-
ple, we took T = 3000K and considered the following choices of parameters:
m
H(0)=1.6·1063 km/sec.mpc and l =1.4·10−33 cm. For this particular case, it
4,3
istobenotedthatonlyforanarrowrangeofinitialconditionscorrespondingtothe
values of c between −0.99145 and −0.99141, outcomes of this specific model are
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within the observed ranges for the cosmological parameters, i.e., Hubble constant
H (60km/sec.mpc−80km/sec.mpc), age t (1.2·1010 yrs−1.6·1010 yrs) and
0 0
density ρ (∼10−31gr/cm3).
0
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