Dark energy and matter from a five-dimensional Chern-Simons cosmology

0.15 MB·English

by  Luis F. Urrutia

Dark energy and matter from a five-dimensional Chern-Simons cosmology∗
Luis F. Urrutia
Departamento de F´ısica de Altas Energ´ıas, Instituto de Ciencias Nucleares
Universidad Nacional Aut´onoma de M´exico, A.P. 70-543, 04510 M´exico D.F., M´exico.
AFriedman-Robertson-Walkercosmologyarisingfromafive-dimensionalChern-Simons(CS)the-
ory for the group S0(1,5) coupled to matter is considered as an alternative model for dark energy
andmatter. Thefour-dimensionalreductiondescribesanacceleratinguniversehavingatimedepen-
dentNewton’scoupling Gandapositive cosmological constant. Fivedimensional mattergives rise
towhat weinterpreteasfour dimensional ordinary plusdark matteranda darkenergy isprovided
by a cosmological constant term plus a fluid arising from the CS coupling. The case of five dust is
studied in detail, leading to acceptable limits for most of the cosmological parameters considered,
in the context of an open non-flat universe. Nevertheless, a value for G˙/G which is two orders of
4
magnitudehigher than recent boundsis predicted.
0
0
2
I. INTRODUCTION
n
a
J The recent use of type I supernovae as standard candles to measure the expansion of the universe has led to the

1 conclusionthatthe universeisaccelerating[1,2,3]. Thisrequiresadarkenergywithsignificantnegativepressure[4].
3 Onepossibilitytoaccountforthisisbasedonarevivalofthecosmologicalconstantcontribution. Alternativewaysof
obtaining negative pressure, in this case time dependent, include a frustrated network of topological defects (such as

1
strings or walls)[5] and an evolving scalar field, named quintessence in some cases [6], for example. The dark energy
v
fractional density is estimated in the range 0.6 < Ω < 0.8, while the corresponding ranges for matter (ordinary
1 DE
plus dark) fractional density are 0.15<Ω <0.45 (open universe) and 0.2<Ω <0.4 (flat universe)[7].
0 m m
0 Inthisworkweproposeanalternativewayofdescribinganaccelerateduniverse,whichisbasedonthecosmological
2 consequences of a five-dimensional Chern-Simons (CS) theory in our four dimensional world.
0 CS gauge actions, defined in odd dimensions, arise as the boundary terms of the Chern classes (F)n, with
4 M2n
F =dA+A A being the curvature two-form associated to the Lie group connection one form A. In fact, one has
/0 that (F)n =d∧Ω2n−1, which defines the CS action to be [8] R
c
q
- SCS(A)= Ω2n−1. (1)
gr ZM2n−1
v: In fieldtheory the lowestdimensionalcase,correspondingto n=2,has been profusely studied in the literature. Two
i of the most relevant examples are: (i) the explicit solution of 2+1 dimensional gravity, rewritten as a CS theory [9]
X
and (ii) the realization of fractional statistics via a three dimensional Abelian CS field [10].
r The next case corresponding to n =3 has received less attention. Following the work in Ref. [11] we start from a
a
five-dimensionalFriedman-Robertson-Walker(FRW)cosmologicalmodelwithmatter. Subsequently,fourdimensional
quantities are introduced, thus providing an observable four dimensional FRW cosmology characterizedby a varying
Newton’ s coupling. The case of vacuum solutions has been previously discussed in Refs. [12, 13]. Five [14] and
higher dimensional [15] cosmologies arise naturally in the study of unified models of the interactions, such as string
theories for example and have been extensively studied in the past [16]. Our chosen five dimensional action includes
the Einstein, the Gauss-Bonet and the cosmological constant terms with precise relative coefficients dictated by the
CS construction. We provide a matter coupling through a five-dimensional fluid.
The paper is organizedas follows: section II summarizes the results of Ref.[11] which are relevant to our purposes.
The five dimensional cosmology is introduced in section III, where the corresponding equations for the scale factors
of the fourth (a(t)) and fifth dimensions (σ(t)) are summarized from the previous work in Refs.[12, 13]. Also, the
four dimensionalNewton ’s coupling andcosmologicalconstantare identified. The restrictionto the case of 5-matter
with zero 5-internal pressure is introduced in section IV and the exact solutions for the scale factors are obtained.
SectionV is devotedto the identificationof the correspondingfour dimensionalquantities whichcanbe subsequently
probed by the observational bounds. Section VI includes a discussion of the five-dust case (corresponding to the
∗ DedicatedtoAlbertoGarc´ıaonhissixthiestbirthday
2
further restrictionof zero 5-externalpressure)with a comparisonwith the observationaldata. Finally, a summary of
the results is given in section VII.
II. THE MODEL
We start from the Chern-Simons action for the gauge groups SO(1,5) and SO(2,4) in five dimensions
3 3
S (A)=k tr A(dA)2+ A3dA+ A5 , (2)
CS
2 5
ZM5 (cid:18) (cid:19)
where the A = Aµ¯B¯C¯JB¯C¯dxµ¯ is the Lie-algebra valued connection one-form. The space-time indices are µ¯ =
(µ, 5), µ = 0,1,2,3, while the Lie-algebra indices are B¯ = (B,6), B = (b,5), b = 0,1,2,3.JB¯C¯ = −JC¯B¯ are the
group generators normalized to tr (JA¯B¯JC¯D¯ JE¯F¯)=ǫA¯B¯C¯D¯E¯F¯. We have omitted the explicit wedge products in (2).
The equations of motion are
ǫA¯B¯C¯D¯C¯D¯E¯F¯FC¯D¯ ∧ FE¯F¯ =0, FA¯B¯ =dAA¯B¯ +AA¯C¯ ∧ AC¯B¯, (3)
wSOhe(r1e,5t)heangdroηuAp¯B¯in=dicdeisaga(r−e,r+ai,s+ed,+o,r+lo,w−e)refodrbSyOt(h2e,4c)o.rrOesupronudniintgs aflraetsmucehtritchsaηtA¯tB¯he=fodrmiagA(−A¯,B¯+,=+,A+µ¯,A¯+B¯,d+x)µ¯foisr
dimensionless.
InordertorecoverEinsteintheoryofgravitywefollowtheworkofRef. [11]andintroducethefollowing(0,1,2,3,5)+
(6) splitting of the connection AB¯C¯ =(A BC, AB6)
ABC =ω˜BC, AB6 =ηe˜B, (4)
where, as we will see from the resulting action, ω˜BC are the five-dimensional Ricci rotation coefficients and e˜B are
the funfbein one-forms. Substituting in (2) one obtains
2
S (ω˜, e˜)=3kη ǫ e˜A R˜BC R˜DE + Σ e˜A e˜B e˜C R˜DE
CS ABCDE
∧ ∧ 3 ∧ ∧ ∧
Z (cid:18)
1
+ Σ2 e˜A e˜B e˜C e˜D e˜E , (5)
5 ∧ ∧ ∧ ∧
(cid:19)
where
R˜AB =dω˜AB+ω˜AC ω˜ B (6)
C
∧
is the five dimensional Riemann tensor. In the above we have defined Σ = λη2, with dimensions 1/L2, where λ is
relatedtothesignatureofthefifthindexgroup. Wehaveλ=+1forSO(2,4)andλ= 1forSO(1,5). We recognize
−
the second term in the RHS of (5) as the five-dimensional Einstein action. The relation
2kηΣ ǫ eA eB eC RDE =12kηΣ R √g d5x, (7)
ABCDE 5 5
∧ ∧ ∧
Z Z
leads to
1
=12kηΣ, (8)
16πG
5
thus allowing the identification of the five-dimensional gravitational constant G . The third term in (5) is the
5
cosmologicalconstant contribution, while the first is the five-dimensional Gauss-Bonet term.
In terms of the new variables (4) the equations of motion (3) are
ǫ SAB SCD =0, ǫ SAB TC =0, (9)
ABCDE ABCDE
∧ ∧
where
SAB =R˜AB+e˜A e˜B, TA =de˜A+ω˜A e˜B. (10)
B
∧ ∧
Here, the torsion TA appears as the curvature R˜A6. In the following we will consider the case of zero torsion.
AfullKaluza-Kleininterpretationofthismodel,wherethefive-dimensionalmetricissplittedinitsfour-dimensional
metric component plus the electromagnetic and scalar fields has been developed in Refs.[13, 17], where non-minimal
couplings to gravity together with non-linear modifications to the standard Einstein-Maxwell-dilaton theory are ob-
tained. Also, plane-fronted gravitationaland electromagnetic waves solutions in spaces endowed with a cosmological
constant term are reported in the first Ref.[17].
3
III. FIVE-DIMENSIONAL COSMOLOGY.
We assume that the action (5) is coupled to general five-dimensional matter described by the generic fluid stress
tensor
T =diag(ρ , p , p , p , p ). (11)
AB 5m 5extm 5extm 5extm 5intm
Also we consider a Fermi-Robertson-Walker (FRW) line element
ds2 = dt2+a2(t)ωiωi+σ2(t)(dx5)2, (12)
−
where we use the standard parameterization for ωi, i = 1,2,3 in the possible cases k = 1, 0. We assume that the
±
fifth coordinate is already compactified. As we will see, this model does not accept σ =cte as a dynamical solution,
so that we will be satisfied with obtaining solutions for σ(t) that decrease in the cosmologicaltime.
After a standardbut tedious calculationwe obtainthe followingequations of motionfor the scale factorsa(t), σ(t)
[12, 13]
2 3
a˙ a˙ σ˙ k 8πG 1 a˙ σ˙ k a˙ σ˙
5
+ + = ρ + Σ, (13)
"(cid:18)a(cid:19) aσ a2# 3 5m− Σ (cid:18)a(cid:19) σ a2aσ!−
σ¨ a¨ σ˙ a˙ k a˙ 2 1 a¨a˙ σ˙ a˙ 2 σ¨ k σ¨
+2 + + + = 8πG p 2 + + 3Σ,
"σ (cid:18)a σa(cid:19) a2 (cid:18)a(cid:19) # − 5 5extm− Σ aaσ (cid:18)a(cid:19) σ a2σ!−
(14)
2 2
a¨ a˙ k 8πG 1 a˙ a¨ k a¨
5
+ + = p + Σ. (15)
"a (cid:18)a(cid:19) a2# − 3 5intm− Σ (cid:18)a(cid:19) a a2a!−
It will prove convenient in the sequel to introduce the following auxiliary variables [12, 13]
2
a¨ σ¨ a˙ k a˙ σ˙
v = +Σ, v = +Σ, v = + +Σ, v = +Σ. (16)
1 a 2 σ 3 a a2 4 aσ
(cid:18) (cid:19)
Using them, Eqs.(13,15,14) are respectively written as
8πG Σ 8πG Σ
5 5
v v = ρ , v v = p ,
3 4 5m 1 3 5intm
3 − 3
2v v +v v = 8πG Σp . (17)
1 4 2 3 5 5extm
−
The terms in square brackets in Eqs. (13,14,15) correspond to the non-zero components G ,G , G , respectively,
00 ii 55
of the five-dimensional Einstein tensor G which satisfy well-known Bianchi identities. These are reflected in the
AB
conservationequationofthe effectivefive-dimensionalenergymomentumtensor,whichisidentifiedbyreadingoffthe
terms in the RHS ofthe aboveequations. In this way we can view the CS cosmologyas an standardfive-dimensional
FRWcosmologywithamodifiedenergymomentumtensorincludingmatter(m),acosmologicalterm(Λ)andacosmic
fluid (D) which will be interpreted as an additional contributionto the dark energy. In other wordswe introduce the
following splitting
ρ =ρ +ρ +ρ , p =p +p +p , (18)
5T 5m 5D 5Λ 5AT 5Am 5AD 5AΛ
where A labels the exterior and interior components of the respective pressures. The individual contributions are
3
3 1 a˙ σ˙ k a˙ σ˙ 3
ρ = + , ρ = Σ,
5D −8πG5 Σ (cid:18)a(cid:19) σ a2aσ! 5Λ −8πG5
2
1 1 a¨a˙ σ˙ a˙ σ¨ k σ¨ 3
p = 2 + + , p = Σ,
5extD 8πG Σ aaσ a σ a2σ 5extΛ 8πG
5 (cid:18) (cid:19) ! 5
3 1 a˙ 2 a¨ k a¨ 3
p = + , p = Σ. (19)
5intD 8πG Σ a a a2a 5intΛ 8πG
5 (cid:18) (cid:19) ! 5
4
The conservation equation for the full effective energy-momentum tensor, which matches the corresponding five-
dimensional Bianchi identity, is
a˙ σ˙
ρ˙ +3 (ρ +p )+ (ρ +p )=0. (20)
5T 5T 5extT 5T 5intT
a σ
We have verified that each cosmic fluid component labeled by Λ or D satisfies the conservation Eq. (20), in such a
way that the matter contribution also does it.
Atthis stagewe comparethe secondandthirdpieces ofthe action(5)with the standardfour-dimensionalEinstein
and cosmological constant terms, respectively,
1
I = √g d4x (R 2Λ ), (21)
4 4 4
16πG −
Z 4
in order to identify the corresponding Newton coupling G and cosmologicalconstant Λ .
4 4
The resulting identification produces
1 2Λ
=12kηΣσ(t)r , 4 =72kηΣ2σ(t)r , (22)
5 5
16πG −16πG
4 4
which leads to
G
G (t)= 5 , Λ = 3Σ= 3λη2, (23)
4 4
r σ(t) − −
5
where r is the compact radius of the fifth dimension. That is to say, S0(1,5), λ = 1 corresponds to the de Sitter
5
−
group, while S0(2,4), λ=+1 corresponds to the anti-de Sitter group. Let us observe that the model predicts a time
dependent Newton coupling G (t) but a cosmologicalconstant Λ . Also we obtain
4 4
G˙ σ˙
4
= . (24)
G −σ
4
We choose to parameterize the equation of state of the five-dimensional matter in the standard way
m
3
p = 1 ρ , p =(M 1)ρ . (25)
5extm 5m 5intm 5m
3 − −
(cid:16) (cid:17)
The conservation equation (20) leads to the general form
1
ρ . (26)
5m ∼ am3σM
The equation of state for the cosmologicalconstant contribution is
p = ρ , (27)
5Λ 5Λ
−
while the equation of state for the D-contribution is dependent on the equations of motion.
The vacuum solution (ρ = 0, p = 0, p = 0) of our model was discussed in Refs. [12, 13]. In the case
5m 5extm 5intm
of a negative deceleration parameter, the unwanted increasing behavior σ(t) = σ¯cosh(Λt) for the scale factor of the
fifth dimension was obtained.
IV. THE CASE OF 5-MATTER WITH 5-EXTERNAL PRESSURE
This corresponds to the case where p =0, (M =1), with ρ =0 and p =0. In order to have a non-zero
5int 5m 5extm
6 6
value for ρ we require both v and v to be simultaneously non-vanishing. This leads to
5 3 4
v =0, a¨+λη2a=0, λ= 1 (28)
1
⇒ ±
and the equation for a(t) is completely decoupled. In the sequel we only consider the case λ= 1, which produces a
−
negative deceleration parameter. The solution is
a(t)=Asinh(ηt). (29)
5
We have
a˙ a¨a
H(t):= =ηcoth(ηt), q(t):= = tanh2(ηt), (30)
a − a˙2 −
together with the relations
η = q(t)H(t) η =√ q H , q(t)<0, (31)
0 0
− ⇒ −
which allows to determine the cosmoplogical constant in terms of the present values of the Hubble and deceleration
parameters.
AsseeninEq. (30)thedecelerationparameterq isnegative,resultthatagreeswithrecentobservationswhichyield
an accelerated expanding universe, characterized by [1, 2]
q = 0.58+0.14, q = 1.0 0.4, (32)
0 − −0.12 0 − ±
respectively.
The next step is to deal with the solution of σ(t). The remaining equations reduce to
8πG Σ m
5 3
v v = ρ , v v = 8πG Σ 1 ρ , (33)
3 4 5m 2 3 5 5m
3 − 3 −
(cid:16) (cid:17)
which ratio leads to the following equation for σ
σ¨ Nηcoth(ηt)σ˙ (1 N)η2σ =0, N =3 m . (34)
3
− − − −
The above can be reduced to a Legendre equation. Imposing appropriate boundary conditions we finally obtain
N
σ =C (x2 1)−ν/2Pν−1(x), ν = 0, x=coth(ηt). (35)
1 − ν 2 ≤
The condition m 3>0 is indeed adequate to a standard matter distribution. The density ρ is given by
3 5m
−
3 k 2ν 1 1
ρ = η2+ . (36)
5m 8πG5 (cid:18) A2(cid:19)Γ(−ν+1))"(x2−1)(ν−3)/2Pνν−1(x)#
The matter density is positive as expected and one can verify that ρ 1/(am3σ), in accordance with the conser-
5m
∼
vation law (20).
V. IDENTIFICATION OF FOUR DIMENSIONAL QUANTITIES
The basic five dimensional equations for our cosmology Eqs.(13,14), together with the definition (18), can be
presented in the form
2 2
a˙ k 8πG (t) a¨ k a˙
4
+ = ρ , 2 + + = 8πG (t)p , (37)
a a2 3 4eff a a2 a − 4 4eff
(cid:18) (cid:19) (cid:18) (cid:19)
in order to identify the four dimensional effective density and pressure. They turn out to be
a˙ 3 1 2 a˙
ρ =r σρ r σ˙ , p =r σp + r σ¨+ r σ˙. (38)
4eff 5 5T 5 4eff 5 5extT 5 5
− a 8πG 8πG 8πG a
5 5 5
From the above equations, together with (15), one can express the five dimensional density and pressures in terms
of the four dimensional effective quantities. Substituting this in the five dimensional conservation equation (20), we
obtainthefourdimensionalconservationequationappropriatetoauniversewithvaryinggravitationalcouplingG (t)
4
a˙ G˙
4
ρ˙ +3 (ρ +p )+ ρ = 0. (39)
4eff 4eff 4eff 4eff
a G
(cid:18) (cid:19) 4
Indeed, equation (39) can also be obtained from the standard FRW conservation after the replacements ρ
4
→
G (t)ρ , p G (t)p are made.
4 4eff 4 4 4eff
→
6
Letus emphasize thatoncethe fourdimensionalinterpretationin enforced,the degreeoffreedomcorrespondingto
the five dimensional scale factor σ(t) is translated into a varying Newton coupling G (t).
4
Nextweprovideafurtherinterpretationoftheindividualpiecesthatmakeupthefourdimensionaleffectivedensity
ρ and the four dimensional effective pressure p . In the cases of matter and cosmological constant we make
4eff 4eff
the choice
ρ =r σρ , p =r σp , (40)
4 5 5 4 5 5ext
which is motivated by the standard reinterpretation of the five-dimensional total matter in a given volume, in terms
of four-dimensional quantities [18, 19, 20]. Also, Eq.(40) preserves the form of the equations of state, leading to
m
3
p = 1 ρ , p = ρ . (41)
4m 4m 4Λ 4Λ
3 − −
(cid:16) (cid:17)
Nevertheless, a difference arises in the individual conservation equations satisfied by each of these components. In
fact, for the cosmologicalconstant we have
3 3
ρ = r σ Σ, p =r σ Σ, (42)
4Λ 5 4Λ 5
− 8πG 8πG
5 5
which satisfies the full conservation equation (39). In the case or matter we recall that we have chosen m =3,M =
3
1,(p = 0), which together with Eq.(26) leads to ρ 1/am3. This implies the standard matter conservation
5intm 4m
∼
equation
a˙
ρ˙ +3 (ρ +p ) = 0, (43)
4,m 4,m 4,m
a
(cid:18) (cid:19)
associated with the first equation of state in (41).
NextweconsidertheremainingcontributiontotheeffectivequantitieslabeledbytheindexD. Herewesimplyread
off the corresponding identification and conservation properties that are necessary to maintain the four dimensional
Bianchi identities induced by the LHS ’s of Eqs.(37). To this end we recall the over all identifications
a˙ σ˙ 3
ρ = ρ +ρ +ρ =r σ ρ +ρ +ρ , (44)
4eff 4m 4D 4Λ 5 5m 5D 5Λ
− aσ8πG
(cid:18) 5(cid:19)
1 σ¨ 2 a˙ σ˙
p = p +p +p =r σ p +p +p + + , (45)
4eff 4m 4D 4Λ 5 5extm 5extD 5extΛ
8πG σ 8πG aσ
(cid:18) 5 5 (cid:19)
together with the full conservation equation (39). Substituting our previous definitions and individual conservation
equations, we are left with the remaining identification
a˙ σ˙ 3
ρ = r σ ρ , (46)
4D 5 5D
− aσ8πG
(cid:18) 5(cid:19)
1 σ¨ 2 a˙ σ˙
p = r σ p + + . (47)
4D 5 5extD
8πG σ 8πG aσ
(cid:18) 5 5 (cid:19)
which must satisfy the conservation equation
a˙ G˙ G˙
ρ˙ +3 (ρ +p )+ ρ = ρ . (48)
4D 4D 4D 4D 4m
a G −G
The above means that matter is a source of the fluid D. The possible consequences of this are not explored in this
work.
Next we turn to the fractional densities. To this end we introduce the ratios Ω among the different densities ρ
4n 4n
and the instantaneous critical density ρ (t). We have
c
ρ (t) 3
Ω (t)= n , ρ (t)= H(t)2, ρ =0.947 10−29g/cm3. (49)
4n C C0
ρ (t) 8πG (t) ×
c 4
From the first Eq.(37) we obtain the usual result
k
1=Ω +Ω +Ω +Ω :=Ω+Ω , Ω = . (50)
4m 4Λ 4D 4k 4k 4k −(aH)2
The fractional density at the present time corresponding to the cosmologicalterm is
η2 Λ
Ω = = = q , (51)
4Λ0 H2 3H2 − 0
0 0
independently of the behavior of σ(t).
7
VI. THE CASE OF FIVE-DUST WITH λ=−1
In order to test some observational consequences of the model under consideration we examine in some detail the
simple case when p =0,(m =3), p =0,(M =1). The solutions are
5extm 3 5intm
a(t)=Asinh(ηt), σ(t)=Bexp( ηt). (52)
−
Let us remark that here we can implement a decreasing fifth scale parameter producing a crack of doom singularity
at t . Using the above solutions together with the first Eq.(17) we obtain
→∞
3 k eηt
ρ = η2+ , (53)
5m 8πG5 (cid:18) A2(cid:19)sinh3(ηt)
where the constant term involving k can be rewritten as
k
=sinh2(ηt )H2(Ω 1), (54)
A2 0 0 0−
and the subindex zero labels the present time. Using the relations (30) together with the four-dimensionalidentifica-
tions described in the previous section it is possible to write all the required fractional densities in terms of Ω and
0
Q := q(t ) as
0 0
−
1
Ω = (Ω Q ) 1+ , Ω =Q ,
4m0 0 0 4Λ0 0
− √Q
(cid:18) 0(cid:19)
1
Ω = (Ω Q ) , Ω =1 Ω . (55)
4D0 0 0 k0 0
− − √Q −
0
Since the individual density contribution of the fluid D is negative for the cases when the matter density is positive,
we choose to interprete it as part of the dark energy density Ω and define
DE0
Ω =Ω +Ω , Ω =Ω +Ω . (56)
DE0 4Λ0 4D0 0 4m0 DE0
In orderto make some numericalestimations with this modelwe consider the followingrangefor the relevantparam-
eters [7]: age of the universe: 12 < t < 18 Gyr, Hubble parameter: 60 < H < 82 km/(segM ), total density
0 0 pc
parameter : 0.85 < Ω < 1.25 [21], and mass density parameter: 0.2 < Ω < 0.4. In the sequel the numerical
0 4m0
quantities will be assumed to have the corresponding units stated above.
For fixed Ω , the first expression in (55) produces a minimum Q for the minimum value Ω = 0.85. This
4m0 0 0
corresponds to a minimum value of t associated to the maximum H = 82. In fact, for a given Q , the model
0 0 0
predicts increasing t ’s for decreasing H ’s, according to Eq. (30).
0 0
ThemodeldisfavorsΩ 1,whichwouldrequireatleastt =21forH =82,withincreasingvaluesforthesmaller
0 0 0
≥
values the Hubble parameter. The maximum values: H = 82 together with t = 18, determine the maximum value
0 0
Q =0.74. For Ω =0.3 this produces the maximum value Ω =0.88 for an open non-flat universe. On the other
0 4m0 0
hand, the minimum value Ω =0.85produces Q =0.71for Ω =0.3,leading to t =17.4for H =82. This same
0 0 4m0 0 0
minimun value Ω = 0.85 with lower values of H leads to increasing ages of the universe, reaching the upper limit
0 0
for H =79. Summarizing, for Ω =0.3 the model predicts an open non-flat universe with 0.85<Ω <0.88, with
0 4m0 0
corresponding 0.55 < Ω < 0.58. Also we have 17.4 < t < 18; 79 < H < 82; and 0.71 < Q < 0.74. The lower
DE0 0 0 0
value Ω = 0.2 is ruled out by the model requiring a minimum of t = 18.3 for Ω = 0.85 with increasing ages
4m0 0 0
for higher Ω . The minimum allowed is Ω = 0.24. Higher values of Ω can be accomadated in the model. For
0 4m0 4m0
example Ω =0.4 requires a minimun age of t =16.8 for H = 82 and Ω =0.85 with Q =0.67. The maximun
4m0 0 0 0 0
age here is obtained with H =76.5. The maximum value for Ω is 0.93.
0 0
Recalling that the model requires a time dependent Newton coupling we now examine the known bounds on this
quantity. In the case of five dust our prediction, according to Eq. (24), is
G˙
4 =+η= Q H 7.0 10−11yr−1. (57)
0 0
G ≈ ×
4
p
for the averages Q = 0.73 and H = 80 determined previously in the case Ω = 0.3. A comparison with one of
0 0 4m0
the latest bounds: 3 10−13yr−1 < G˙ /G <4 10−13yr−1 [22], shows that unfortunately the predicted value
4 4
− × 0 ×
is too high by almost two orders of m(cid:16)agnitud(cid:17)e. The range of allowed values of either Q0 or H0 will not allow any
drastic modification of the estimation (57) in the case of five-dust.
8
VII. SUMMARY
AFRWcosmologicalmodelarisingfromafive-dimensionalChern-SimonstheoryforthegroupS0(1,5)isconsidered.
The resulting five dimensional equations for the corresponding scale parameters a(t) and σ(t) are exactly solved in
the case of five-matter with 5-external pressure. The four-dimensional reduction describes an accelerating universe
(q <0)havingatimedependentNewton’scouplingG (t)andapositivecosmologicalconstant. Thefivedimensional
0 4
mattergivesrisetowhatweinterpreteasfourdimensionalordinaryplusdarkmatterΩ andadarkenergyΩ =
4m0 DE0
Ω Ω is providedby the cosmologicalconstantplus afluid componentarisingfromthe CScoupling. The caseof
0 4m0
−
five dust (zero 5-external pressure) is studied in more detail, leading to a decreasing behavior of the fifth dimension
scale parameter, reaching zero for infinite cosmic time. For the choices Ω = 0.3 and 79 < H < 82 the dust
4m0 0
model predicts an open non flat universe with values: 0.85<Ω <0.88, 17.4<t <18Gyr, and 0.71< q <0.74.
0 0 0
Unfortunately, a value for G˙ /G which is two orders of magnitude higher than recent bounds is found. −
4 4
Within the framework presented so far it remains to be examined whether the choice p =0, (m =3), could
5extm 3
6 6
alleviate some of the above problems. Finally, we have the open alternative to explore solutions with p = 0,
5intm
6
(M = 1), which is not considered at all in this paper. The improvement of the five-dust model over the vacuum
6
situation provides some degree of optimism in this respect.
[1] S.Perlmutteret.al.,Astrophys.J.517(1999)565;S.Perlmutter,M.S.TurnerandM.White,Phys.Rev.Lett.83(1999)670.
[2] A.G. Riess et. al.,Astron. J. 116(1998)1009.
[3] For a detailed review see for example: N.A. Bahcall, J.P. Ostriker, S. Perlmutter and P.J. Steinhardt, Science
284(1999)1481; W.L. Freedman and M.S. Turner, Rev.Mod. Phys.75(2003)1433.
[4] J.E. Peebles, Astrophys. J. 284(1984)439: J.P. Ostriker and P.J. Steinhardt,Nature 377(1995)600.
[5] A.Vilenkin, Phys.Rev.Lett. 53(1984)1016: D. Spergel and U.-L- Pen, Astrophys.J. 491(1997)L67.
[6] Seefor example: J. A.Frieman, C.T. Hill, A.Stebbinsand I. Waga, Phys. Rev.Lett.75(1995)2077; R.Caldwell, R. Dave
and P. Steinhardt,Phys. Rev.Lett.80(1998)1582; I. Zlatev, L. Wangand P. Steinhardt,Phys.Rev. Lett.82(1999)896.
[7] K.Hagiwara et. al.,Phys.Rev.D66(2002)01001-1.
[8] For a review of CS theories, see for example D. Birmingham, M. Blau, M. Rakowsky and G. Thompson, Phys.
Rep.209(1991)129.
[9] E. Witten, Nucl.Phys. B311(1988)46.
[10] For a review, see for example A. Lerda, Anyons, quantum mechanics of particles with fractional statistics, Lecture Notes
in Physics, Vol.m14, Springer-Verlag, 1992.
[11] A.H.Chamseddine, Phys. Letts. B233(1989)291.
[12] A. Mac´ıas, H. Morales-T´ecotl, N. Morales and L.F. Urrutia, in VIII J.A. Swieca Summer School on Particles and Fields,
Eds. J. Barcelos-Neto, S.F. Novaesand V.O. Rivelles, World Scientific, Singapore, 1996, p. 502.
[13] N. Morales, Algunos problemas relacionados con las Teor´ıas de Chern-Simons, Ph.D. Thesis, Universidad Aut´onoma
Metropolitana, M´exico, May 2000.
[14] A.ChodosandS.Detweiler,Phys.Rev.D21 (1980)2167; R.A.MatznerandA.Mezzacappa,Phys.Rev.D32(1985)3114;
R. A. Matzner and A. Mezzacappa, Found. of Phys. 16(1986)227; M. Rosenbaum, M. Ryan, L. Urrutia and R. Matzner,
Phys.Rev.D36(1987)1032.
[15] P.G.O. Freund,Nucl. Phys.B209(1982)146; C. Wetterich, Phys.Letts. 113B(1982)377; O. Shafi and C. Wetterich,Phys.
Letts. 129B(1983)387; M. Gleiser, S. Rajpoot and J.G. Taylor, Phys. Rev. D30(1984)756; D. Lorentz-Petzold, Phys.
Letts. 153B(1985)134; B.C. Paul and S. Mukherjee, Phys. Rev. D42(1990)2595.
[16] For a review see for example: Modern Kaluza-Klein Theories, ed. by T.A. Appelquist, A. Chodos and P.G.O. Freund,
Addison-WesleyPublishing Company,Redwood City, California.
[17] A.Mac´ıas and E. Lozano, Phys. Rev. D67(2003)085009; ibid. Mod. Phys. Letts. A16(2001)2421.
[18] E. Kolb, D.Lindley and D. Seckel,Phys. Rev. D30(1984)1205.
[19] L. Farin˜a-Busto, Phys. Rev. D38(1988)1741.
[20] K.Kleidis and D . Papadopoulos, Gen. Rel. Grav 29(1997)275.
[21] A.Melchiorri et. al., Astrophys. J. 536(2003)L63.
[22] C. J. Copi, A. N.Davis and L. M. Krauss, A new Nucleosyntthesis Constraint on the Variation of G, astro-ph/0311334.