Reconstruction of Five-dimensional Bounce cosmological Models From Deceleration Factor

0.12 MB·English

by  Lixin Xu

TP-DUT/2006-1
Reconstruction of Five-dimensional Bounce cosmological Models From Deceleration
Factor
Lixin Xu,∗ Hongya Liu†,‡ and Yongli Ping
Department of Physics, Dalian University of Technology, Dalian, 116024, P. R. China
Inthispaper,weconsideraclassoffive-dimensionalRicci-flatvacuumsolutions,whichcontaintwo
arbitraryfunctionsµ(t)andν(t). Itisshownthatµ(t)canberewrittenasanewarbitraryfunction
f(z) in terms of redshift z and the f(z) can be determined by choosing particular deceleration
parameters q(z) which gives early deceleration and late time acceleration. In this way, the 5D
cosmological model can bereconstructed and the evolution of theuniverse can be determined.
6 PACSnumbers: 04.50.+h,98.80.-k.
0 Keywords: Kaluza-Kleintheory;cosmology
0
2

n
a I. INTRODUCTION
J

0
Recent observations of high redshift Type Ia supernovae reveal that our universe is undergoing an accelerated
2
expansion rather than decelerated expansion [1, 2, 3]. In addition, the discovery of Cosmic Microwave Background
1 (CMB) anisotropy on degree scales together with the galaxy redshift surveys indicate Ωtotal ≃ 1 [4] and Ωm ≃ 1/3.
v All these resultsstronglysuggestthat the universeis permeatedsmoothly by ’darkenergy’,whichviolatesthe strong
1 energy condition with negative pressure. The dark energy and accelerating universe has been discussed extensively
7
from different points of view [5, 6, 8]. In principle, a natural candidate to dark energy could be a small cosmological
4
constant. However, there exist serious theoretical problems: fine tuning problem and coincidence problem. To
1
overcome the fine tuning problem, some self-interact scaler fields φ with an equation of state (EOS) w = p /ρ
0 φ φ φ
6 were introduced dubbed quintessence, where wφ is time varying and negative. In nature, the potentials of the scalar
0 field would be determined from the underlying physical theory, such as Supergravity, Superstring/M-theory etc..
/ However, disregarding these underlying physical theories just from the phenomenal level, we can design many kinds
h
of potentials to solve the concrete problems [7]. Once the potentials are given, EOS w of dark energy can be found.
p φ
- On the contrary, the potential can be reconstructed by a given EOS wφ [9]. Then, the forms of the scalar potential
o
are obtained by the observations data with given EOS w . However, it is mentioned that the same models have
φ
r
t early deceleration epoch. And, the acceleration of the universe just begins at not distance past. So, starting from
s
the decelerationparameter,we can also reconstructthe cosmologicalmodels [10]. This is the main motivation of this
a
: paper.
v
The idea that our world may have more than four dimensions is due to Kaluza [11], who unified Einstein’s theory
i
X of General Relativity with Maxwell’s theory of Electromagnetism in a 5D manifold. In 1926, Klein reconsidered the
r idea and treated the extra dimension as a compact small circle topologically [12]. However, up to now, there is no
a experimental evidence and indication for the existence of extra dimensions. Nevertheless, there is strong theoretical
motivation for considering our world spacetime with more than three spatial dimension. M-theory is a higher di-
mensional theory with seven extra spatial dimensions, which try to incorporate quantum gravity in a consistent way.
Recently, in higher dimension frame, brane world model has been studied extensively. In brane world model, our
four dimensional world is a hypersurface (brane) embedded in a higher dimensional manifold (bulk) and all forces
are confined on the brane but gravity can propagate in the bulk. Also, it has interesting consequence in cosmology,
for recent review seen [13]. Also, in higher dimensional frame, the Space-Time-Matter (STM) theory as a modern
Kaluza-Klein theory is designed to incorporate the geometry and matter by Wesson and his collaborators [14]. In
STMtheory,ourworldishypersurfaceembeddedinfive-dimensionalRicciflat(R =0)manifoldandallthematter
AB
inour worldareinducedfromthe higherdimension. The theory requiresthe metric componentscontaining the extra
dimension, saying y. And, the compactification or not of the extra dimension is not necessary in STM theory, for
† Correspondingauthor
∗Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
2
Ricci flat (R = 0) five-dimensional manifold. The consequent cosmology in STM theory is studied in [15], [16],
AB
[17].
II. DARK ENERGY IN A CLASS OF FIVE-DIMENSIONAL COSMOLOGICAL MODEL
Within the framework of STM theory, a class of exact 5D cosmological solution was given by Liu and Mashhoon
in 1995 [19]. Then, in 2001, Liu and Wesson [15] restudied the solution and showed that it describes a cosmological
model with a big bounce as opposed to a big bang. The 5D metric of this solution reads
dr2
dS2 =B2dt2−A2 +r2dΩ2 −dy2 (1)
(cid:18)1−kr2 (cid:19)
where dΩ2 ≡ dθ2+sin2θdφ2 and
(cid:0) (cid:1)
ν2+K
A2 = µ2+k y2+2νy+ ,
µ2+k
(cid:0) (cid:1)
1∂A A˙
B = ≡ . (2)
µ ∂t µ
Here µ = µ(t) and ν = ν(t) are two arbitrary functions of t, k is the 3D curvature index (k=±1,0), and K is a
constant. This solution satisfies the 5D vacuum equation R =0. So, the three invariants are
AB
I ≡ R=0,I ≡RABR =0,
1 2 AB
72K2
I = R RABCD = . (3)
3 ABCD A8
The invariant I in Eq. (3) shows that K determines the curvature of the 5D manifold. It wouldbe pointed out that
3
the 5D and4D Planckmassarenotrelateddirectly,because of5G =0,inthe STMtheory. So,in4D we cantake
AB
κ2 =8πG , where G is 4D Newtonian gravitation constant.
4 4 4
Using the 4D part of the 5D metric (1) to calculate the 4D Einstein tensor, one obtains
3 µ2+k
(4)G0 = ,
0 (cid:0) A2 (cid:1)
2µµ˙ µ2+k
(4)G1 = (4)G2 =(4) G3 = + . (4)
1 2 3 AA˙ A2
In our previous works [17], the induced matter was set to be a conventionalmatter plus a time variable cosmological
‘constant’ or three components: dark matter radiation and x-matter. In [18], we pointed out the correspondence
between the arbitrary function f(z) and scalar field potentials. In this paper, we assume that the induced matter
contains two parts, for simplicity: cold dark matter (CDM) ρ and dark energy (DE) ρ . So, we have
cd de
3 µ2+k
= ρ +ρ ,
(cid:0) A2 (cid:1) cd de
2µµ˙ µ2+k
+ = −(p +p ), (5)
AA˙ A2 cd de
where
p =0, p =w ρ . (6)
cd de de de
From Eqs.(5) and (6), one obtains the EOS of the dark energy
p 2µµ˙/AA˙ + µ2+k A2
w = de =− , (7)
de ρ 3(µ2+k)/A(cid:0)2−ρ (cid:1)(cid:14)A−3
de cd0
and the dimensionless density parameters
ρ ρ
cd cd0
Ω = = , (8)
cd ρ +ρ 3(µ2+k)A
cd de
Ω = 1−Ω . (9)
de cd
3
where ρ = ρ¯ A3, and Ω and Ω are dimensionless density parameters of CDM and DE respectively. The
cd0 cd0 0 cd de
Hubble parameter and deceleration parameter should be given as [15], [17],
A˙ µ
H ≡ = (10)
AB A
d2A dA 2 Aµ˙
q(t,y) ≡ −A =− , (11)
dτ2(cid:30)(cid:18)dτ (cid:19) µA˙
from which we see that µ˙/µ >0 represents an accelerating universe, µ˙/µ <0 represents a decelerating universe. So
the function µ(t) playsa crucialroleofdefining the propertiesofthe universeat late time. Inthis paper,we consider
the spatially flat k =0 cosmological model. From the above Eqs. (7)-(11), it is easy to find that the equations does
not contain ν(t) explicitly, which is included in A. So, to avoid boring choice of the concrete forms ν(t), we use
A /A =1+z anddefineµ2 µ2 =f(z), andthenfindthatthe Eqs. (7)-(11)canbe reducedtoasfollowsinredshift
0 0 z
z (cid:14)
1+(1+z)dlnf(z)/dz
w = − , (12)
x 3−3Ω
cd
Ω = Ω (1+z)f(z), (13)
cd cd0
Ω = 1−Ω , (14)
de cd
1+3Ω w (1+z)dlnf(z)
q = x x =− . (15)
2 2 dz
From Eq. (15), we find that it is an ordinary differential equation of function f(z) w.r.t. redshift z, once one
form of q(z) is given. In Ref. [10], the scalar potentials are constructed from a given deceleration parameter q =
−1−pap/(1+ap), where p is a constant. In our case, with this spirit, we also can reconstruct the forms of function
f(z) from a given concrete form of q(z). As a example, we consider the following deceleration parameter
1 α
q(z)= − , (16)
2 (1+z)β
where, α and β are positive constants determined by observation values. By this assumption, one obtains f(z)
Cexp −2α(1+z)−β
β
f(z)= (cid:16) (cid:17), (17)
1+z
where, C = exp(2α/β) is an integral constant determined by f(0) ≡ 1. Once f(z) is specified, the evolution
parameters, such as Ω Ω and w are obtained at one time. From Eq. (16), one fine that
cd de x
1
q = −α. (18)
0
2
Thecurrentobservationvalueq =−0.67±0.25determinesthe parameterα. And,the parameterβ hasrelationwith
0
the ratio of Ω and Ω , which is constrained by early cosmological observations. By set α =1.17 and β =1.7, the
cd de
evolution of these parameters is plotted in Fig.
III. CONCLUSIONS
Ageneralclassof5Dcosmologicalmodelsischaracterizedbyabigbounceasopposedtothebigbangin4Dstandard
cosmological model. This exact solution contains two arbitrary functions µ(t) and ν(t). By careful observation, one
findsthatthedimensionlessdensity,decelerationparametersandEOSofdarkenergyareindependentonν(t)explicitly
whichisincludedinA. Intermsofredshifz,µ(t)canberewrittenintoanewarbitraryfunctionf(z). Oncetheforms
ofthearbitraryfunctionf(z)arespecified,theuniverseevolutionwillbedetermined. Inthispaper,wereconstructthe
arbitraryfunctionf(z)byassumingaparticularformofdecelerationparameterq(z). Inthisway,the5Dcosmological
models are reconstructed and the evolution of the universe can be determined.
4
1
W
cd
0.75
0.5 q
0.25
W
0 de
-0.25 w
x
-0.5
-0.75
2.5 5 7.5 10 12.5 15 17.5 20
z
FIG.1: TheevolutionofthedimensionlessdensityparametersΩcd,Ωde,anddecelerationparameterq,EOSofdarkenergywx
versusredshift z,where Ωcd0 =0.25, Ωde0 =0.75, α=1.17 and β=1.7.
Acknowledgments
This work was supported by NSF (10273004)and NBRP (2003CB716300)of P. R. China.
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