E. Papantonopoulos- Brane Cosmology

8/3/2019 E. Papantonopoulos- Brane Cosmology

1/25

a r X i v : h e p - t h / 0 2 0 2 0 4 4 v 2 2 2 F e b 2 0 0 2

NTUA022002

hep-th/0202044

Brane Cosmology

E. Papantonopoulos a

National Technical University of Athens, Physics Department, ZografouCampus, GR 157 80, Athens, Greece.

Abstract

The aim of these lectures is to give a brief introduction to branecosmology. After introducing some basic geometrical notions, we dis-cuss the cosmology of a brane universe with matter localized on thebrane. Then we introduce an intrinsic curvature scalar term in thebulk action, and analyze the cosmology of this induced gravity. Fi-

nally we present the cosmology of a moving brane in the backgroundof other branes, and as a particular example, we discuss the cosmo-logical evolution of a test brane moving in a background of a Type-0string theory.

Lectures presented at the First Aegean Summer School on Cosmology,Samos, September 2001.

a e-mail address:[email protected]
http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044v2http://arxiv.org/abs/hep-th/0202044http://arxiv.org/abs/hep-th/0202044http://arxiv.org/abs/hep-th/0202044v2


2/25

1 Introduction

Cosmology today is an active eld of physical thought and of exiting exper-imental results. Its main goal is to describe the evolution of our universefrom some initial time to its present form. One of its outstanding successesis the precise and detailed description of the very early stages of the universeevolution. Various experimental results conrmed that ination describesaccurately these early stages of the evolution. Cosmology can also help tounderstand the large scale structure of our universe as it is viewed today.It can provide convincing arguments why our universe is accelerating and itcan explain the anisotropies of the Cosmic Microwave Background data.

The mathematical description of Cosmology is provided by the Einstein

equations. A basic ingredient of all cosmological models is the matter con-tent of the theory. Matter enters Einstein equations through the energymomentum tensor. The form of the energy momentum tensor depends onthe underlying theory. If the underlying theory is a Gauge Theory, the scalarsector of the theory must be specied and in particular its scalar potential.Nevertheless, most of the successful inationary models, which rely on ascalar potential, are not the result of an uderlying Gauge Theory, but ratherthe scalar content is arbitrary xed by hand.

In String Theory the Einstein equations are part of the theory but thetheory itself is consistent only in higher than four dimensions. Then the cos-mological evolution of our universe is studied using the effective four dimen-sional String Theory. In this theory the only scalar available is the dilatoneld. The dilaton eld appears only through its kinetic term while a dilatonpotential is not allowed. To have a dilaton potential with all its cosmologicaladvantages, we must consider a corrections to the String Theory. Becauseof this String Theory is very restrictive to its cosmological applications.

The introduction of branes into cosmology offered another novel approachto our understanding of the Universe and of its evolution. It was proposed[1] that our observable universe is a three dimensional surface (domain wall, brane ) embedded in a higher dimensional space. In an earlier speculation,motivated by the long standing hierarchy problem, it was proposed [ 2] that

the fundamental Planck scale could be close to the gauge unication scale,at the price of large spatial dimensions, the introduction of which explainsthe observed weakness of gravity at long distances. In a similar scenario [ 3],our observed world is embedded in a ve-dimensional bulk, which is stronglycurved. This allows the extra dimension not to be very large, and we can

1


3/25


4/25

!

5

Figure 2: A vector moving around a curve C .

of this new cosmological set-up, and more importantly to discuss the math-

ematical tools necessary for the construction of a brane cosmological model.For this reason will not exhaust all the aspects of brane cosmology. For us, tosimplify things, a brane is a three dimensional surface which is embedded in ahigher dimensional space which has only one extra dimension, parametrizedby the coordinate y, as it is shown in Fig.1.

The lectures are organized as follows. In section two after an elementarygeometrical description of the extrinsic curvature, we will describe in somedetail the way we embed a D-dimensional surface in a D+1-dimensional bulk.We believe that understanding this procedure is crucial for being able toconstruct a brane cosmological model. In section three we will present theEinstein equations on the brane and we will solve them for matter localized

on the brane. Then we will discuss the Friedmann-like equation we get onthe brane and the ways we can recover the standard cosmology. In sectionfour we will see what kind of cosmology we get if we introduce in the bulkaction a four-dimensional curvature scalar. In section ve we will consider abrane moving in the gravitational eld of other branes, and we will discussthe cosmological evolution of a test brane moving in the background of atype-0 string theory. Finally in the last section we will summarize the basicideas and results of brane cosmology.

2 A Surface embedded in a D-dimensionalManifold M

3


5/25

2.1 Elementary Geometry

To understand the procedure of the embedding of a surface in a higherdimensional manifold, we need the notion of the extrinsic curvature . Westart by explaining the notion of the curvature [ 5].

If you have a plane P and a curve C on it, then any vector startingfrom the point p and moving along the curve C it will come at the samepoint p with the same direction as it started as shown in Fig.2. However,if the surface was a three dimensional sphere, and a vector starting from pis moving along C , will come back on p, having different direction from thedirection it started with, as shown in Fig.3.

These two examples give the notion of the curvature . The two dimen-

sional surface is at, while the three dimensional surface is curved. Havingin mind Fig.2 and Fig.3 we can say that a space is curved if and only if someinitially parallel geodesics fail to remain parallel. We remind to the readerthat a geodesic is a curve whose tangent is parallel-transported along itself,that is a straightest possible curve.

5

Figure 3: A vector moving around a curve C on a sphere.

We can dene the notion of the parallel transport of a vector along a

curve C with a tangent vector ta , using the derivative operator a . A vectorva given at each point on the curve is said to be parallely transported, as onemoves along the curve, if the equation

ta a vb (1)

4


6/25

is satised along the curve. Consider a point q on with a normal vector

na

. If I parallel transport the vector na

to a point p, then it will be thedashed lines. The failure of this vector to coincide with the vector na at p corresponds intuitively to the bending of in the space time in which isembedded (Fig.4). This is expressed by the extrinsic curvature

K ab = hca cnb (2)

where hab the metric on .

6

5

.

.

Figure 4: The notion of extrinsic curvature.

2.2 The Embedding Procedure

Imagine now that we have a surface (Known also as a domain wall orbrane) embedded in a D-dimensional Manifold M [6]. Assume that M splitsin two parts M . We demand the metric to be continuous everywhere andthe derivatives of the metric to be continuous everywhere except on . TheEinstein-Hilbert action on M is

S EH = 12 dD xgR (3)

where gMN is the metric on M , with M,N=1,...,D. We dene the inducedmetric on as

hMN = gMN nM nN (4)5


7/25

where nM is the unit normal vector into M . We vary the action ( 3) in M and we get

S EH = 12 dD 1hgMN n p( M gNP P gMN ) (5)

If we replace gMN = hMN + nM nN in (5) we get

S EH = 12 dD 1hh MN n p( M gNP P gMN ) (6)

Recognize the term ( M gNP P gMN ) as the discontinuity of the deriva-tive of the metric across the surface. We do not like discontinuities so weintroduce on both side of the surface , the Gibbons-Hawking boundary term

S GH = 12 dD 1hK (7)

where k = hMN K MN and K MN is the extrinsic curvature dened in ( 2). If we vary the Gibbons-Hawking term we get as expected

S GH = 12 dD 1h(K 12Kh MN gMN ) (8)

we need the variation of K . If we use the variation

nM =12nM n

P n

QgP Q (9)

after some work we get

K = K MN g hMN nP ( M gNP P gMN ) +12

Kn P nQ gP Q (10)

We have the variation of both terms S EH and S GH in (6) and (8). Putingeverything together we get

S EH + S GH = dD 1xh 12hMN nP M gNP + K MN gMN (11) 12Kn M nN gMN 12Kh MN gMN

To simplify the above formula imagine a vector X M tangential to , thenM X M = hMN M X N + nM nN M X N (12)

6


8/25


9/25

to be zero, using (17) and (19) we get the Israel Matching Conditions [ 7]

K MN Kh MN = T MN (21)where the curly brackets denote summation over both sides of . This re-lation is central for constructing any brane cosmological model. In someway it supplements the Einstein Equations in such a way as to make themconsistent on the brane.

3 Brane Cosmology in 5-dimensional Space-time

3.1 The Einstein Equations on the Brane

The most general action describing a three-dimensional brane in a ve-dimensional spacetime is [8]

S (5) =1

2k25 M d5xg (5) R 25 + 12k24 d4xg (4) R 24+ M d5xgLmat5 + d4xgLmat4 (22)

where 5 and 4 are the cosmological constants of the bulk and brane re-spectively, and Lmat5 , Lmat4 are their matter content. From the dimensionfulconstants k25 , k24 the Planck masses M 5, M 4 are dened as

k25 = 8 G (5) = M 35k24 = 8 G (4) = M 24 (23)

We will derive the Einstein equations for the simplied action

S (5) =1

2k2(5) M d5xg(5) R + M d5xgLmat5 (24)which is historically the rst action considered [ 9], and we will leave for laterthe general case. We will consider a metric of the form

ds2 = gMN dxM dxN = g dxdx + b2dy2 (25)

8


10/25

where y paramertizes the fth dimension. We will assume that our four

dimensional surface is sited at y = 0. We allow a time dependence of theelds so our metric becomes

ds2 = n2(t, y)dt2 + a2(t, y)ij dxi dx j + b2(t, y)dy2 (26)Note that we take for simplicity at metric for the ordinary spatial di-

mensions. For the matter content of the action ( 24) we assume that matteris conned on both, brane and bulk. Then, the energy momentum tensorderived from (24) can be decomposed into

T M N = T M

N bulk+ T M N brane (27)

For the matter on the brane we consider perfect uid with

T M N brane =(y)

bdiag (,p,p,p,, 0) (28)

What we want now is to study the dynamics of the metric g (t, 0). For this,we have to solve the 5-dimensional Einstein equations

GMN = k25T MN (29)

Inserting ( 26) in (29) we get

G00 = 3aa

(aa

+bb

) n2

b2a

a+

a

a(a

a b

b)

Gij =a2

b2ij

a

a(a

a+ 2

n

n)

b

b(n

n+ 2

a

a) + 2

a

a+

n

n

+a

n2ij

aa

(aa

+ 2nn

) +bb

(2aa

+nn

) 2aa

bb

G05 = 3(n n

aa

+a

abb

a

a)

G55 = 3a

a(a

a+

n

n)

b2

n2aa

(aa

nn

) +aa

(30)

We are looking for solutions of the Einstein equations ( 29) near or inthe vicinity of y = 0. At the point y = 0, where the brane is situated,we must take under consideration the Israel Boundary Conditions. We canuse relations (21) to calculate them or follow an easier way [9]. We require

9


11/25

that the derivatives of the metric with respect to y, to be discontinuous at

y = 0. This means that in the second derivatives of the quantities a and n adistributional term will appear which will have the form [ a

](y) or [n

](y)with

[a

] = a

(0+ ) a

(0) (31)[n

] = n

(0+ ) n

(0) (32)We can calculate the quantities ( 31) and (32) using equations ( 30) and

the energy-momentum tensor ( 28)

[a

]a0b0

= k2(5)3

[n

]n0b0

= k2(5)

3(3 p + 2 ) (33)

where a0 = a(t, 0) and b0 = b(t, 0) and we have set n(t,0)=1. If we use areection symmetry y y the (55) component of the Einstein equations(29) with the use of (28) and (33) becomes

a20a20

+a0a0

= k4(5)36

( + 3 p) k2(5)T 553b20

(34)

This is our cosmological Einstein equation which governs the cosmological

evolution of our brane universe .

3.2 Cosmology on the Brane

Dene the Hubble parameter from H = a 0a 0 . Then equation ( 34) becomes

2H 2 + H = k4(5)36

( + 3 p) k2(5)T 553b20

(35)

If one compares equation ( 35) with usual Friedmann equation, one can seethat energy density enters the equation quadratically, in contrast with theusual linear dependence. Another novel feature of equation ( 35) is that thecosmological evolution depends on the ve-dimensional Newtons constantand not on the brane Newtons constant.

10


12/25

To have a feeling of what kind of cosmological evolution we get, we con-

sider the Bianchi identity M GM

N = 0. Then using the Einstein equation(29) and the energy momentum tensor on the brane from ( 28), we get

+ 3( + p)a0a0

= 0 (36)

which is the usual energy density conservation. If we take for the equation of state p = w, then the above equation gives for the energy density the usualrelation

a0e3(1+ w ) (37)If we look for power law solutions however, a0(t) tq equation ( 35) gives

q =1

3(1 + w)(38)

which comparing with qstandard = 23(1+ w ) gives slower expansion.If we add a cosmological constant in the bulk, then a solution of the

Einstein equation ( 29) can be obtained, in which the universe starts witha non conventional phase and then enters the standard cosmological phase[10, 11, 12].

4 Induced Gravity on the BraneThe effective Einstein equations on the brane which we discussed in section(3.1) were generalized in [13], where matter conned on the brane was takenunder consideration. However, a more fundamental description of the physicsthat produces the brane could include [ 14] higher order terms in a derivativeexpansion of the effective action, such as a term for the scalar curvature of the brane, and higher powers of curvature tensors on the brane. In [ 15, 16] itwas observed that the localized matter elds on the brane (which couple tobulk gravitons) can generate via quantum loops a localized four-dimensionalworldvolume kinetic term for gravitons. That is to say, four-dimensional

gravity is induced from the bulk gravity to the brane worldvolume by thematter elds conned to the brane. We will therefore include the scalarcurvature term in our action and we will discuss what is the effect of thisterm to cosmology.

11


13/25

Our theory is described then by the full action ( 22). Using relations ( 23)

we dene a distance scaler c

2524

=M 24M 35

. (39)

Varying ( 22) with respect to the bulk metric gMN , we obtain the equations(5) GMN = 5gMN + 25 ((5) T MN + (loc )T MN (y)) , (40)

where(loc )T MN

124

(4) g(5) g (4) GAB 24 (4) T MN + 4gMN (41)is the localized energy-momentum tensor of the brane. (5) GMN , (4) GMN denote the Einstein tensors constructed from the bulk and the brane metricsrespectively. Clearly, (4) GMN acts as an additional source term for the branethrough (loc )T MN .

It is obvious that the additional source term on the brane will modify theIsrael Boundary Conditions ( 21). The modied conditions are

[K ] = 25 bo (loc )T (loc )T

3 , (42)

where the bracket means discontinuity of the extrinsic curvature K =12b yg across y = 0, and bo = b(y = 0). A Z 2 symmetry on reectionaround the brane is understood throughout.

Using equations (40) and (42) we can derive the four-dimensional Einsteinequations on the brane [ 8]. They are

(4) G = k2(4)

(4) T (4) a L +L2

+32

a (43)

where 2/r c, while the quantities L are related to the matter contentof the theory through the equationL L

L2

4 = T

14

32 + 2 T , (44)

and L L . The quantities T

are given by the expression

T = 4 12

5 24 (4) T + (45)+

23

25(5) T +

(5) T yy (5) T

4 E

. (46)

12


14/25

Bars over (5) T and the electric part E

= C M N nM nN of the Weyl tensor

C M NP R mean that the quantities are evaluated at y = 0.

E

carries theinuence of non-local gravitational degrees of freedom in the bulk onto thebrane and makes the brane equations ( 43) not to be, in general, closed. Thismeans that there are bulk degrees of freedom which cannot be predicted fromdata available on the brane. One needs to solve the eld equations in thebulk in order to determine E

on the brane.

4.1 Cosmology on the Brane with a (4)R TermTo get a feeling of what kind of cosmology we get on the brane with an (4) Rterm, we consider the metric of ( 26). To simplify things we take (5) T MN to be just the ve-dimensional cosmological constant, while for the matterlocalized on the brane we take (4) T MN to have the usual form of a perfectuid (relation( 28)). The new term that enters here in the calculations is thefour-dimensional Einstein tensor (4) GMN which appears in (loc ) T (relation(41)). The non vanishing components of (4) GMN can be calculated to be

(4) G00 = 3(y)k24b

a2

a2+

n2

a2

(4) Gij = (y)k24b

a2

n2ij

a2

a2+ 2

anan 2

aa ij (47)

Then as in section (3.1) we can calculate the distributional parts of thesecond derivatives as [17]

[a

]a0b0

= k2

3 +

k25k24n20

a02

a20+

n20a20

[n

]n0b0

=k2

3(3 p + 2 ) +

k25k24n20

a20a20 2

a0n0a0n0

+ 2a0a0

n20a20

(48)

If we again use a reection symmetry y y, then the Einstein equa-tions (40) with the use of (28), (47) and (48) give our cosmological Einsteinequation

H 2 2k24k25 H 2 + 1a20 = k23 + 1a20 (49)

13


15/25

To compare this equation with the evolution equation ( 34) we had derived

without the(4)

R term, we observe that the energy density enters the evolu-tion equation linearly as in the standard cosmology. However the evolutionequation ( 49) is not the standard Friedmann cosmological equation. We canrecover the usual Friedmann equation [ 18], if (neglecting the 1a 20 term)

H 1M 2(4)2M 3(5)

(50)

If we use the crossover scale r c of (39) the above relation means that anobserver on the brane will see correct Newtonian gravity at Hubble distancesshorter than a certain crossover scale, despite the fact that gravity propagates

in extra space which was assumed there to be at with innite extent; atlarger distances, the force becomes higher-dimensional. We can get the samepicture if we look at the equation ( 44). If the crossover scale r c is large,then 2/r c is small and the last term in ( 44) decouples, giving the usualfour-dimensional Einstein equations.

5 A brane on a Move

So far the domain walls (branes) were static solutions of the underlying the-ory, and the cosmological evolution of our universe was due mainly to the

time evolution of energy density on the domain wall (brane). In this sectionwe will consider another approach. The cosmological evolution of our uni-verse is due to the motion of our brane-world in the background gravitationaleld of the bulk [6, 19, 20, 21].

In [6] the motion of a domain wall (brane) in a higher dimensional space-time was studied. The Israel matching conditions were used to relate thebulk to the domain wall (brane) metric, and some interesting cosmologicalsolutions were found. In [20] a universe three-brane is considered in motion inten-dimensional space in the presence of a gravitational eld of other branes.It was shown that this motion in ambient space induces cosmological ex-pansion (or contraction) on our universe, simulating various kinds of matter.

In particular, a D-brane moving in a generic static, spherically symmetricbackground was considered. As the brane moves in a geodesic, the inducedworld-volume metric becomes a function of time, so there is a cosmologicalevolution from the brane point of view. The metric of a three-dimensionalbrane is parametrized as

14


16/25


17/25

This equation is the standard form of a at expanding universe. If we

dene the scale factor as 2

= g then we can calculate the Hubble constantH = , where dot stands for derivative with respect to cosmic time. Thenwe can interpret the quantity ( )

2 as an effective matter density on the branewith the result

83

ef f =(C + E )2gS e2 |g00|(gS g3 + 2e2)

4|g00|grr gS g3(gg

)2 (60)

Therefore the motion of a three-dimensional brane on a general sphericallysymmetric background had induced on the brane a matter density. As it isobvious from the above relation, the specic form of the background willdetermine the cosmological evolution on the brane.

We will go to a particular background, that of a Type-0 string, and seewhat cosmology we get. The action of the Type-0 string is given by [ 23]

S 10 = d10xg e R + ( )2 14( T )2 14m2T 2 112H mnr H mnr

12

(1 + T +T 2

2)|F 5|2 (61)

The equations of motion which result from this action are

2 2

4( n )2

1

2m2T 2 = 0 (62)

Rmn +2 m n 14 m

T n T 1

4 4!e2f (T ) F mklpq F n klpq

110

Gmn F sklpq F sklpq = 0 (63)

( 2 + 2 n n + m2)T +1

2 5!e2f (T )F sklpq F sklpq = 0 (64)

m f (T )F mnkpq = 0 (65)

The tachyon is coupled to the RR eld through the function

f (T ) = 1 + T +12

T 2 (66)

16


18/25

In the background where the tachyon eld acquires vacuum expectation value

T vac = 1, the tachyon function ( 66) takes the value f (T vac ) =12 whichguarantee the stability of the theory [ 24].

The equations ( 62)-(65) can be solved using the metric ( 51). Moreoverone can nd the electrically charged three-brane if the following ansatz forthe RR eld

C 0123 = A(r ), F 0123r = A(r ) (67)and a constant value for the dilaton eld = 0 is used

g00 =

H

12 , g(r ) = H

12 , gS (r ) = H

12 r 2, grr (r ) = H

12 , H = 1 +

e0 Q

2r 4(68)

5.1 Cosmology of the Moving Brane

The induced metric on the brane ( 56) using the background solution ( 68) is

ds2 = ( H 12 + H

12 r 2 + H

12 r 2hij i j )dt2 + H

12 (dx )2 (69)

From equation ( 65) the RR eld C = C 0123 using the ansatz ( 67) becomes

C

= 2 Qg2g52

s grr f 1(T ) (70)

where Q is a constant. Using again the solution ( 68) the RR eld can beintegrated to give

C = e0 f 1(T )(1 + e0 Q2r 4

)1 + Q1 (71)

where Q1 is a constant. The effective density on the brane ( 60), using equa-tion (68) and (70) becomes [22]

83

ef f =14

[(f 1(T ) + EH e 0 )2 (1 +2e20

2H )]

Q2e20

r 10H

52 (72)

where the constant Q1 was absorbed in a redenition of the parameter E .

Identifying g = 2 and using g = H 1

2 we get from (72)83

ef f = (2e0

Q)

12 f 1(T ) + Ee

0

42

1 +2e20

6(2e0

Q)

12

(1 4)12 (1 4)

52 (73)

17


19/25

From the relation g = H 12 we nd

r = ( 4

1 4)

14 (

Qe0

2)

14 (74)

This relation restricts the range of to 0 < 1, while the range of r is0 r < . We can calculate the scalar curvature of the four-dimensionaluniverse asRbrane = 8 (4 + )ef f (75)

If we use the effective density of (73) it is easy to see that Rbrane of (75) blowsup at = 0. On the contrary if r 0, then the ds210 of (51) becomes

ds210 = r2

L (dt2 + ( dx )2) + Lr 2 dr 2 + Ld5 (76)with L = ( e

0 Q2 )

12 . This space is a regular AdS 5 S 5 space.Therefore the brane develops an initial singularity as it reaches r = 0,

which is a coordinate singularity and otherwise a regular point of the AdS 5space. This is another example in Mirage Cosmology [20] where we canunderstand the initial singularity as the point where the description of ourtheory breaks down.

If we take 2 = 0, set the function f (T ) to each minimum value and alsotaking 0 = 0, the effective density ( 73) becomes

83

ef f = (2Q

)12 (2 +

E 4

)2 1 (1 4)52 (77)

As we can see in the above relation, there is a constant term, coming fromthe tachyon function f (T ). For small and for some range of the parametersE and Q it gives an inationary phase to the brane cosmological evolution.In Fig.5 we have plotted ef f as a function of for Q = 2. Note here thatE is constrained from ( 57) as C + E 0. In our case using (71) we getE 24, therefore E can be as small as we want.The cosmological evolution of a brane universe according to this exampleis as follows. As the brane moves away from r = 0 to larger values of r ,the universe after the inationary phase enters a radiation dominated epochbecause the term 4 takes over in (77). As the cosmic time elapses the 8term dominates and nally when the brane is far away from r = 0, the termwhich is controlled by the angular momentum 2 gives the main contribution

18


20/25


21/25

parameter scales like the square of the energy density and this results in a

slower universe expansion. There were a lot of extensions and modicationsof this model, trying to get the standard cosmology but it seems that theuniverse in a brane world passed from an unconvensional phase at its earlieststages of its cosmological evolution.

The inclusion of an (4) R term in the action, offered a more natural expla-nation of the brane unconvensional phase. At small cosmological distancesour universe was involved according the usual Einstein equations. If the cos-mological scale is larger than a crossover scale, we enter a higher-dimensionalregime where the cosmological evolution of our brane universe is no longercoverned by the conventional Friedmann equation.

We also presented a model where a brane is moving in the gravitationaleld of other branes. Then we can have the standard cosmological evolutionon the brane, with the price to be paid, that the matter on the brane is amirage matter.

References

[1] T. Regge and C. Teitelboim, Marcel Grossman Meeting on General Rel-ativity, Trieste 1975, North Holland; V. A. Rubakov and M. E. Sha-poshnikov, Do we live inside a domain wall? Phys. Lett. B 125 (1983)136 .

[2] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, The hierarchy problem and new dimensions at a millimeter, Phys. Lett. B 429 (1998) 263 [hep-ph/9803315 ]; Phenomenology, astrophysics and cosmology of theorieswith submillimeter dimensions and TeV scale quantum gravity, Phys.Rev. D 59 (1999) 086004 [hep-ph/9807344 ];

I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, New di-mensions at a millimeter to a Fermi and superstrings at a TeV, Phys.Lett. B 436 (1998) 257 [hep-ph/9804398 ].

[3] R. Sundrum, Effective eld theory for a three-brane universe, Phys. Rev.D 59 (1999) 085009 [hep-ph/9805471 ]; Compactication for a threebrane universe, Phys. Rev. D 59 (1999) 085010 [hep-ph/9807348 ];

L. Randall and R. Sundrum, Out of this world supersymmetry breaking,Nucl. Phys. B.557 (1999) 79 [hep-th/9810155 ]; A large mass hierarchy

20
http://arxiv.org/abs/hep-ph/9803315http://arxiv.org/abs/hep-ph/9803315http://arxiv.org/abs/hep-ph/9807344http://arxiv.org/abs/hep-ph/9804398http://arxiv.org/abs/hep-ph/9805471http://arxiv.org/abs/hep-ph/9807348http://arxiv.org/abs/hep-th/9810155http://arxiv.org/abs/hep-th/9810155http://arxiv.org/abs/hep-ph/9807348http://arxiv.org/abs/hep-ph/9805471http://arxiv.org/abs/hep-ph/9804398http://arxiv.org/abs/hep-ph/9807344http://arxiv.org/abs/hep-ph/9803315http://arxiv.org/abs/hep-ph/9803315


22/25

from a small extra dimension, Phys. Rev. Lett. 83 (1999) 3370 [hep-

ph/9905221 ].[4] J. Polchinski, Dirichlet branes and Ramond-Ramond charges, Phys. Rev.

Lett. 75 (1995) 4724 [hep-th/9510017 ];

P. Horava and E. Witten, Heterotic and type-I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209 ].

[5] R. M. Wald, General Relativity , University of Chicago Press (1984).

[6] H. A. Chamblin and H. S. Reall, Dynamic dilatonic domain walls, hep-th/9903225 ; A. Chamblin, M. J. Perry and H. S. Reall, Non-BPS D8-

branes and dynamic domain walls in massive IIA supergravities, J.High Energy Phys. 09 (1999) 014 [hep-th/9908047 ].

[7] W. Israel, Singular hypersurfaces and thin shells in general relativity,Nuovo Cimento 44B [Series 10] (1966) 1; Errata-ibid 48B [Series 10](1967) 463 ;

G. Darmois, M emorial des sciences math ematiques XXV (1927) ;

K. Lanczos, Untersuching uber achenhafte verteiliung der materie in der Einsteinschen gravitationstheorie (1922) , unpublished; Fl achenhafteverteiliung der materie in der Einsteinschen gravitationstheorie, Ann.Phys. (Leipzig) 74 (1924) 518 .

[8] G. Konas, General brane cosmology with (4) R term in (A)dS 5 or Minkowski bulk, JHEP 0108 (2001) 034, [hep-th/0108013 ].

[9] P. Binetruy, C. Deffayet and D. Langlois, Nonconventional cosmology from a brane universe, [hep-th/9905012 ].

[10] P. Binetruy, C. Deffayet, U. Ellwanger and D. Langlois, Brane cosmo-logical evolution in a bulk with cosmological constant, Phys. Lett. 285 B477 (2000), [hep-th/9910219 ].

[11] N. Kaloper and A. Linde, Ination and large internal dimensions, Phys.Rev. D 59 (1999) 101303 [hep-th/9811141 ];

N. Arkani-Hamed, S. Dimopoulos, N. Kaloper and J. March-Russell,Rapid asymmetric ination and early cosmology in theories with submil-limeter dimensions, [hep-ph/9903224 ];

21
http://arxiv.org/abs/hep-ph/9905221http://arxiv.org/abs/hep-ph/9905221http://arxiv.org/abs/hep-th/9510017http://arxiv.org/abs/hep-th/9510209http://arxiv.org/abs/hep-th/9903225http://arxiv.org/abs/hep-th/9903225http://arxiv.org/abs/hep-th/9908047http://arxiv.org/abs/hep-th/0108013http://arxiv.org/abs/hep-th/9905012http://arxiv.org/abs/hep-th/9910219http://arxiv.org/abs/hep-th/9811141http://arxiv.org/abs/hep-ph/9903224http://arxiv.org/abs/hep-ph/9903224http://arxiv.org/abs/hep-th/9811141http://arxiv.org/abs/hep-th/9910219http://arxiv.org/abs/hep-th/9905012http://arxiv.org/abs/hep-th/0108013http://arxiv.org/abs/hep-th/9908047http://arxiv.org/abs/hep-th/9903225http://arxiv.org/abs/hep-th/9903225http://arxiv.org/abs/hep-th/9510209http://arxiv.org/abs/hep-th/9510017http://arxiv.org/abs/hep-ph/9905221http://arxiv.org/abs/hep-ph/9905221


23/25

N. Arkani-Hamed, S. Dimopoulos, G. Dvali and N. Kaloper, Innitely

large new dimensions, [hep-th/9907209 ];R. N. Mohapatra, A. Perez-Lorenzana, C. A. de S. Pires,Int.J.Mod.Phys. A16 (2001) 1431, [hep-ph/0003328 ];N. Kaloper. Bent domain walls as brane-worlds, Phys. Rev. D60 (1999)123506 [hep-th/9905210 ];

P. Kanti, I. I. Kogan, K. A. Olive and M. Pospelov, Cosmological 3-Brane Solution, Phys. Lett. B468 (1999) 31 [hep-ph/9909481 ].

[12] G. Dvali and S. H. H. Tye, Brane ination, Phys. Lett. B450 (1999) 72 [hep-ph/9812483 ];

E. E. Flanagan, S. H. H. Tye and I. Wasserman, A cosmology of thebrane world, [hep-ph/9909373 ];

H. B. Kim and H. D. Kim, Ination and Gauge Hierarchy in Randall-Sundrum Compactication, [hep-th/9909053 ];

C. Csaki, M. Graesser, C. Kolda and J. Terning, Cosmology of OneExtra Dimension with Localized Gravity, Phys. Lett. B462 (1999) 34[hep-ph/9906513 ];

J. Cline, C. Grojean and G. Servant, Cosmological Expansion in thePresence of Extra Dimensions, Phys. Rev. Lett. 83 (1999) 4245 , [hep-

ph/9906523 ];C. Csaki, M. Graesser, L. Randall and J. Terning, Cosmology of BraneModels with Radion Stabilization, [hep-ph/9911406 ].

[13] T. Shiromizu, K. Maeda and M. Sasaki, The Einstein equations on the3-Brane World, Phys. Rev. D62 (2000) 024012 , [gr-qc/9910076].

[14] R. Sundrum, Effective eld theory for a three-brane universe, Phys. Rev.D59 (1999) 085009 , [hep-ph/9805471 ].

[15] G. Dvali, G. Gabadadze and M. Porati, 4D Gravity on a Brane in 5D Minkowski Space, Phys. Lett. B485 (2000) 208 , [hep-th/0005016 ].

[16] G. Dvali and G. Gabadadze, Gravity on a Brane in Innite VolumeExtra Space, Phys. Rev. D63 ,065007 (2001),[hep-th/0008054 ].

22
http://arxiv.org/abs/hep-th/9907209http://arxiv.org/abs/hep-ph/0003328http://arxiv.org/abs/hep-th/9905210http://arxiv.org/abs/hep-ph/9909481http://arxiv.org/abs/hep-ph/9812483http://arxiv.org/abs/hep-ph/9909373http://arxiv.org/abs/hep-th/9909053http://arxiv.org/abs/hep-ph/9906513http://arxiv.org/abs/hep-ph/9906523http://arxiv.org/abs/hep-ph/9906523http://arxiv.org/abs/hep-ph/9911406http://arxiv.org/abs/gr-qc/9910076http://arxiv.org/abs/hep-ph/9805471http://arxiv.org/abs/hep-th/0005016http://arxiv.org/abs/hep-th/0008054http://arxiv.org/abs/hep-th/0008054http://arxiv.org/abs/hep-th/0005016http://arxiv.org/abs/hep-ph/9805471http://arxiv.org/abs/gr-qc/9910076http://arxiv.org/abs/hep-ph/9911406http://arxiv.org/abs/hep-ph/9906523http://arxiv.org/abs/hep-ph/9906523http://arxiv.org/abs/hep-ph/9906513http://arxiv.org/abs/hep-th/9909053http://arxiv.org/abs/hep-ph/9909373http://arxiv.org/abs/hep-ph/9812483http://arxiv.org/abs/hep-ph/9909481http://arxiv.org/abs/hep-th/9905210http://arxiv.org/abs/hep-ph/0003328http://arxiv.org/abs/hep-th/9907209


24/25

[17] C. Deffayet,Cosmology on a brane in Minkowski bulk, Phys. Lett. B502

(2001) 199 ,[hep-th/0010186 ].[18] R. Dick, Brane worlds, Class. Quantum Grav. 18 (2001) R1, [hep-

th/0105320 ].

[19] P. Kraus, Dynamics of Anti-de Sitter Domain Walls, JHEP 9912:011(1999) [hep-th/9910149 ].

[20] A. Kehagias and E. Kiritsis, Mirage cosmology, JHEP 9911:022 (1999)[hep-th/9910174 ].

[21] E. Papantonopoulos, Talk given at IX Marcel Grossman meeting, Rome

July 2000, To appear in the proccedings .[22] E. Papantonopoulos and I. Pappa, Type 0 Brane Ination from Mirage

Cosmology, Mod. Phys. Lett. A 15 (2000) 2145 [hep-th/0001183 ].

[23] I. R. Klebanov and A. A. Tseytlin, D-Branes and Dual Gauge Theoriesin Type 0 Strings, Nucl.Phys. B 546 (1999) 155 [hep-th/9811035 ].

[24] I. R. Klebanov Tachyon Stabilization in the AdS/CFT Correspondence[hep-th/9906220 ].

[25] I. R. Klebanov and A. Tseytlin, Asymptotic Freedom and Infrared Be-havior in the type 0 String Approach to gauge theory, Nucl.Phys. B 547(1999) 143 [hep-th/9812089 ].

[26] J. A. Minahan, Asymptotic Freedom and Connement from Type 0 String Theory, JHEP 9904:007 (1999) [hep-th/9902074 ].

[27] E. Papantonopoulos and I. Pappa, Cosmological Evolution of a BraneUniverse in a Type 0 String Background Phys. Rev. D63 (2001) 103506 [hep-th/0010014 ].

[28] J. Y. Kim, Dilaton-driven brane ination in type IIB string theory , [hep-

th/0004155 ].[29] J. Y. Kim, Brane ination in tachyonic and non-tachyonic type OB

string theories [hep-th/0009175 ].

23
http://arxiv.org/abs/hep-th/0010186http://arxiv.org/abs/hep-th/0105320http://arxiv.org/abs/hep-th/0105320http://arxiv.org/abs/hep-th/9910149http://arxiv.org/abs/hep-th/9910174http://arxiv.org/abs/hep-th/0001183http://arxiv.org/abs/hep-th/9811035http://arxiv.org/abs/hep-th/9906220http://arxiv.org/abs/hep-th/9812089http://arxiv.org/abs/hep-th/9902074http://arxiv.org/abs/hep-th/0010014http://arxiv.org/abs/hep-th/0004155http://arxiv.org/abs/hep-th/0004155http://arxiv.org/abs/hep-th/0009175http://arxiv.org/abs/hep-th/0009175http://arxiv.org/abs/hep-th/0004155http://arxiv.org/abs/hep-th/0004155http://arxiv.org/abs/hep-th/0010014http://arxiv.org/abs/hep-th/9902074http://arxiv.org/abs/hep-th/9812089http://arxiv.org/abs/hep-th/9906220http://arxiv.org/abs/hep-th/9811035http://arxiv.org/abs/hep-th/0001183http://arxiv.org/abs/hep-th/9910174http://arxiv.org/abs/hep-th/9910149http://arxiv.org/abs/hep-th/0105320http://arxiv.org/abs/hep-th/0105320http://arxiv.org/abs/hep-th/0010186


25/25

[30] D. Youm, Brane Ination in the Background of a D-Brane with NS

B eld , [hep-th/0011024 ]; Closed Universe in Mirage Cosmology , [hep-th/0011290 ].

24
http://arxiv.org/abs/hep-th/0011024http://arxiv.org/abs/hep-th/0011290http://arxiv.org/abs/hep-th/0011290http://arxiv.org/abs/hep-th/0011290http://arxiv.org/abs/hep-th/0011290http://arxiv.org/abs/hep-th/0011024

E. Papantonopoulos- Brane Cosmology

Documents

Transcript of E. Papantonopoulos- Brane Cosmology