$5D$ Solutions to $Λ$CDM Universe Derived from Global Brane Model

0.14 MB·English

by  Yongli Ping

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ModernPhysicsLetters A
(cid:13)c WorldScientificPublishingCompany
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0 5D Solutions to ΛCDM Universe Derived from Global Brane Model
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n
a
J YongliPing,LixinXu,BaorongChang,MolinLiuandHongyaLiu
0 School of Physics and Optoelectronic Technology,
2 Dalian Universityof Technology, Dalian, Liaoning 116024, P.R.China
[email protected]

]
h
t Received
-
Revised
p
e
h Anexact solution of brane universe is studied and the result indicates that Friedmann
[ equations onthebranearemodifiedwithanextraterm.Thistermcanplaytheroleof

dark energy and make the universe accelerate. In order to derive the ΛCDM Universe
1
fromthisglobalbranemodel,thenewsolutionsareobtainedtodescribethe5Dmanifold.
v
3 Keywords:ΛCDM;brane;cosmology.
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0 PACSnumbers:04.50.+h,98.80.-k,02.40.-k
3
.
1 1. Introduction
0
8 Recent observations indicate that our universe is undergoing accelerated
0 expansion1,2 and dominated by a negative pressure component dubbed dark en-
:
v ergy.Obviously,anaturalcandidateto darkenergyis a cosmologicalconstantwith
i
X equationofstatewΛ =−1.Einstein(1917)introducedthecosmologicalconstantΛ,
3
r becausehebelievedthattheuniverseisstatic. However,Friedmann(1922)discov-
a
ered an expanding solution to the Einstein field equations in the absence of Λ and
Hubble (1929)foundthe universewasexpanding.4,5 Soonafter,Einsteindiscarded
the cosmologicalconstant and admitted his greatest blunder. Although abandoned
by Einstein, the cosmologicalconstantstagedseveralcome-backs.It was soonreal-
ized that, since the static Einstein universe is unstable to small perturbations, one
could construct expanding universe models which had a quasi-static origin in the
past, thus ameliorating the initial singularity which plagues expanding FRW mod-
els. Theoretical interest in Λ remained on the increase during the 1970s and early
1980s with the construction of inflationary models, in which matter (in the form
of a false vacuum, as vacuum polarization or as a minimally coupled scalar-field)
behavedprecisely like a weakly time-dependent Λ-term.The cosmologicalconstant
6
makes an important appearance in models with spontaneous symmetry breaking.
The current interest in Λ stems mainly comes from observations of Type Ia high
redshift supernovae which indicate that the universe is accelerating expansion fu-
eledperhapsbyasmallcosmologicalΛ-term.1,2 Thereviewaboutthecosmological
1
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2 Y. Ping et al.
constant can be seen.6,7,8,9
It is proposed that our universe is a 3-brane embedded in a higher-dimensional
space.10,11,12,13,14,15,16,17 In brane-world model, gravity can freely propagate
in all dimensions, while standard matter particles and forces are confined on the
3-brane.A five-dimensional(5D) cosmologicalmodeland derivedFriedmannequa-
18
tions on the branes are considered by Binetruy, Deffayet and Langlois (BDL),
for a recent review, it can be seen.19,20 Brane-world models of dark energy are
21
studied and accelerating universe comes from gravity leaking to extra dimension
22
in DGP brane.
In this paper, we derive ΛCDM universe from globalbrane model with a Ricci-
flat bulk characterized by a class of exact solutions. The solutions were firstly pre-
sented by Liu and Mashhoon and restudied latter by Liu and Wesson.23,24 And
these solutions are algebraically rich because they contain two arbitrary functions
of time t. The solutions are utilized in cosmology25,26,27,28,29,30,31,32,33 and are
relate to the brane model.34,35,36 In order to induce ΛCDM universe from global
brane model, more exact solutions of the 5D bulk are obtained.
2. Friedmann equations in global brane universes
23
A class of 5D Ricci-flat cosmologicalsolution reads
dr2
dS2 =B2dt2−A2 +r2dΩ2 −dy2, (1)
1−kr2
(cid:18) (cid:19)
ν2+K
A2 = µ2+k y2+2νy+ , (2)
µ2+k
(cid:0) (cid:1)
1∂A A˙
B = ≡ , (3)
µ ∂t µ
where dΩ2 =dθ2+sin2θdψ2; µ=µ(t) and ν =ν(t) are two arbitrary functions of
time t; k is the 3D curvature index (k = ±1,0), and K is a constant. Because the
5D manifold (1)-(3) is Ricci-flat, we have I ≡R=0, I ≡RABR =0, and
1 2 AB
72K2
I ≡RABCDR = , (4)
3 ABCD A8
soK isrelatedtothe5Dcurvature.TheyareusedasthebulksolutionsoftheBDL-
type brane model. To obtainbrane models for using the Z reflectionsymmetry on
2
36
A and B, they are set as
ν2+K
A2 = µ2+k y2−2ν |y |+ ,
µ2+k
(cid:0)1∂A (cid:1)A˙
B = ≡ . (5)
µ ∂t µ
Then the corresponding 5D bulk Einstein equations are taken as
G = κ2 T ,
AB (5) AB
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5D Solutions toΛCDM Universe Derived from Global Brane Model 3
TA = δ(y)diag(ρ ,−p ,−p ,−p ,0)
B 1 1 1 1
+δ(y−y )diag(ρ ,−p ,−p ,−p ,0) (6)
2 2 2 2 2
where the first brane is at y =y =0 and the second is at y =y >0. In the bulk
1 2
T = 0 and G = 0, Eq.(6) are satisfied by (5). On the branes, Liu had solved
AB AB
Eq.(6) in Ref. 36. We adopt the result at y =y =0 and y =y >0 as follows:
1 2
6ν
κ2 ρ = , (7)
(5) 1 A2
1
2 ∂ ν 4ν
κ2 p = − ( )− , (8)
(5) 1 A˙1∂t A1 A21
and
6 µ2+k ν
κ2 ρ = ( y − ), (9)
(5) 2 A A 2 A
2 2 2
2 ∂ µ2+k ν
κ2 p = − ( y − )
(5) 2 A˙2∂t A2 2 A2
4 µ2+k ν
− ( y − ), (10)
2
A A A
2 2 2
where, A is the scale factor on y = y = 0 brane and A is the scale factor on
1 1 2
y =y >0 brane.
2
Now, we consider the universe on the second brane, i.e. y = y > 0. From the
2
5D metric (1), the Hubble anddecelerationparameterson the y =y brane canbe
2
defined as
1 A˙ µ
2
H (t,y)≡ = , (11)
2
B A A
2 2 2
A µ˙
2
q (t,y)= − . (12)
2 µA˙
2
Substituting Eq.(11) into Eq.(9) to eliminate µ2, Eq. (9) can be rewritten into a
new form as
k κ2 6 ν
H2+ = (5)(ρ + ). (13)
2 A2 6y 2 κ2 A2
2 2 (5) 2
We define ρ = 6 ν and it can play the role of dark energy. Then from the
x κ2 A2
(5) 2
Eq.(10), we have
2µµ˙ µ2+k κ2
(5)
+ =− (p −p ), (14)
A2A˙2 A22 2y2 2 x
where p =− 2 ( ν˙ + ν ). Meanwhile, the conservation law TB =0 gives
x κ2(5) A2A˙2 A22 A;B
A˙
2
ρ˙ +3(ρ +p ) =0. (15)
2 2 2
A
2
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4 Y. Ping et al.
FromEqs.(13)and(14),itcanbeseenthattheextratermwhichwillbetreatedas
37
darkenergyhavebeeninducedonthe brane. Byassumingthatonlydarkmatter
is containedonthe brane,we havep =0 and ρ =ρ A3 A3.Then, fromEq.(13)
2 2 20 20 2
and Eq. (14), for k =0,EOS of dark energy,dimensionless density parametersand
deceleration parameters q with A =1 and ν =1 can be obtained
2 20 0
p 1 A ν˙
x 2
w = =− ( +1), (16)
x ρx 3 A˙2ν
1
Ω = , (17)
2
1+νA (1−Ω )
2 20
Ω =1−Ω , (18)
x 2
1 ν˙ ν 1−Ω
20
q = − + +1 , (19)
2 2(cid:20) (cid:18)A2A˙2 A22(cid:19)1+νA2(1−Ω20) (cid:21)
where Ω is current value of matter density parameter Ω . If the function ν is
20 2
given,theevolutionsofallcosmicobservableparametersin(16)-(19)aredetermined
uniquely.
3. Solutions to ΛCDM universe derived from global brane
The Friedmann equations in four-dimensional ΛCDM universe are
k κ2 Λ
H2+ = (4)ρ+ , (20)
a2 3 3
a¨ κ2 Λ
(4)
=− (ρ+3p)+ , (21)
a 6 3
wherethereisonlythematteri.eρ=ρ andp=p =0.Meanwhile,the equation
m m
of state on the cosmological constant Λ is w = −1. In order to get the LCDM
Λ
universe from the global brane, from Eq. (16), we find when
p =−ρ , (22)
x x
the property on the brane tends to ΛCDM universe. And, we can find κ2 =
(4)
κ2 /(2y ). Since κ2 =M−3 and κ2 =M−2, the relation of the four dimensional
(5) 2 (5) (5) (4) (4)
Planck mass is expressed with five dimensional Planck mass as
M2 =2y M3 . (23)
(4) 2 (5)
Therefore, the four dimensional Planck mass is relevant to five dimensional Planck
mass and the position of brane.
From Eq. (22), on the second brane i.e. y =y 6=0 we have
2
A ν˙
2
=2. (24)
A˙ ν
2
So, the relation of ν and A is
2
ν =CA2, (25)
2
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5D Solutions toΛCDM Universe Derived from Global Brane Model 5
where C is a integral constant. We can eliminate the arbitrary function ν in Eq.
(5). Therefore, the (5) is written as
C2A4+K
A2 = µ2+k y2−2CA2 |y |+ 2 . (26)
2 2 2 2 µ2+k
(cid:0) (cid:1)
And this equation is rewritten as
1 2 1 1 K
A2− (2C|y |+1)(µ2+k) = (|y |+ )(µ2+k)2− . (27)
2 2C2 2 C4 2 4 C2
(cid:20) (cid:21)
Therefore, the solutions of this equation are
1 1 1 K
A2 = (2C|y |+1)(µ2+k)± (|y |+ )(µ2+k)2− , (28)
2 2C2 2 C4 2 4 C2
r
where for A2 ≥ 1 (2C|y |+1)(µ2+k), “+” sign is taken; while for A2 ≤
2 2C2 2 2
1 (2C|y |+1)(µ2+k), we choose “−” sign. This scale factor gives a clearer ge-
2C2 2
ometrical description about the 5D spacetime. For K = 0, we can find I = I =
1 2
I =0 and5D bulk is a5-dimensionalflatspacetime.So,the scalefactorinthe 5D
3
flat spacetime is
1 1 1
A2 = C|y |+ ± |y |+ (µ2+k). (29)
2 C2 2 2 2 4
r !
For a flat 3D space i.e. k =0, in the 5D flat spacetime, Eq. (29) is simplified as
1 1 1
A2 = C|y |+ ± |y |+ µ2. (30)
2 C2 2 2 2 4
r !
Using the red-shift relation
A
20
A = , (31)
2
1+z
from the Eq.(30), we have
1 1 1 A2
C|y |+ ± |y |+ µ2 = 20 . (32)
C2 2 2 2 4 (1+z)2
r !
So, for different y , the A is different. In other words, we obtain different A in
2 20 20
different brane. Therefore, if considering when z = 0, A = 1 on different brane,
20
the redshift z should be redefined on different brane.
In fact, these two solutions all can induce the ΛCDM universe on y 6= 0 brane.
C is an arbitrary constant. Substituting ν = CA2 into Eq. (19), the deceleration
2
parameters is rewritten as
1 3C(1−Ω )
20
q = − +1 . (33)
2
2 1+CA (1−Ω )
(cid:20) 2 20 (cid:21)
The present value of deceleration parameters q is
2
1 3C(1−Ω )
20
q = − +1 . (34)
20
2 1+C(1−Ω )
(cid:20) 20 (cid:21)
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6 Y. Ping et al.
Our universe is accelerating, so the deceleration parameter is q < 0. Therefore,
20
the rangeof C is C >1/(2−2Ω ) orC <−1/(1−Ω ). Adopting q =−0.5and
20 20 20
Ω =0.3, we have C =20/7≈3.
20
If we utilize the result ν = CA2 on y = 0 brane i.e ν = CA2, the density on
2 1
y = 0 brane will be ρ = 6C/κ2 and p = −6C/κ2 . Therefore, the y = 0 brane
1 (5) 1 (5)
is dominated by abnormal matter with w =−1.
4. Conclusions
In this paper, the exact globalsolutions ofbrane universesare discussed. The solu-
tionscontaintwoarbitraryfunctionsµandνoftimet.Onthebrane,theFriedmann
equationsaremodifiedbytheextratermwithν.Therefore,thearbitraryfunctionν
will influence the evolutionof our universe.Then we find dimensionless density pa-
rametersanddecelerationparametersarerelatetothearbitraryfunctionν.ΛCDM
universeis the mostsimple modelandnotruledoutbypresentastronomicalobser-
vation. But we do not know where it comes from. Now, in order to derive ΛCDM
universe from the 5D spacetime, the arbitrary function ν is eliminate in the scale
factorofglobalbrane.Sothemoreexactsolutionsoftheglobalbraneareobtained.
Then a clear 5D manifold is presented.
Acknowledgments
This work was supported by NSF (10573003), NSF (10647110), NSF (10703001),
NBRP (2003CB716300)of P. R. China and DUT 893321.
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