Is brane cosmology predictable?

Gen. Relativ. Gravit. (2005) 37(8): 1419–1425DOI 10.1007/s10714-005-0126-8

LETTER

Shin’ichi Nojiri · Sergei D. Odintsov

Is brane cosmology predictable?

Received: 9 December 2004 / Published online: 16 August 2005C© Springer-Verlag 2005

Abstract The creation of the inflationary brane universe in 5d bulk Einstein andEinstein-Gauss-Bonnet gravity is considered. We demonstrate that the emerginguniverse is ambiguous due to arbitrary function dependence of the junction con-ditions (or freedom in the choice of boundary terms). We argue that some funda-mental physical principle (which may be related with AdS/CFT correspondence)is necessary in order to fix the 4d geometry in a unique way.

Keywords Einstein-Gauss-Bonnet gravity · Brane model · Inflationarycosmology

Despite the number of the efforts the construction of the theory of all fundamentalinteractions is still very far from the end. At the moment (super)string/M-theoryremains to be the most promising candidate of unified theory. String/M-theorylives in higher dimensional space from which the early 4d universe naturallyemerges in brane-world approach [1]. Brane cosmology (for recent review andlist of references, see [2,3]) is somehow different from usual 4d cosmology dueto the presence of extra terms having the higher dimensional origin. Nevertheless,there appeared the arguments that early brane universe may be quite a realisticpossibility.

It remains not completely clear for all followers of brane-worlds that branecosmology is not predictable without some additional physical principle. Thejunction conditions (or cosmological equations) are ambiguous (up to the arbitrary

S. NojiriDepartment of Applied Physics, National Defence Academy, Hashirimizu,Yokosuka 239-8686, JapanE-mail: [email protected], [email protected]

S. D. Odintsov (B)Institucio Catalana de Recerca i Estudis Avancats (ICREA) and Institut d’Estudis Espacials deCatalunya (IEEC), Edifici Nexus, Gran Capita 2-4, 08034 Barcelona, SpainE-mail: [email protected]

1420 S. Nojiri, S. D. Odintsov

function) and do not define the four-dimensional geometry in the unique way. Inthe present letter we demonstrate this explicitly for two models: 5d Einstein and5d Einstein-Gauss-Bonnet (EGB) gravity. As a brane universe the inflationary (de-Sitter) space is considered. It is shown that creation of dS instanton always occursjust by corresponding choice of junction conditions (or by the choice of surfaceterms). The predictability of brane cosmology may be achieved only when fun-damental physical principle (for instance, AdS/CFT correspondence) is applied tofix the surface action.

Let us start from 5d Einstein gravity where two (identified with each other)bulk spacetimes are glued at the 4d surface (boundary). The starting action S isthe sum of the Einstein-Hilbert action SEH, the Gibbons-Hawking surface termSGH [4] and the surface action S1:

S = SEH + SGH + 2S1 (1)

SEH = 1

16πG

∫d5x

√g

(R(5) + 12

l2

)(2)

SGH = 1

8πG

∫d4x

√h∇µnµ. (3)

Here nµ is the unit vector perpendicular to the surface, with direction of nµ tobe inside the bulk. The induced metric is denoted as hµν = gµν − nµnν . TheEuclidean signature is used but changing g → −g and h → −h the Lorentziansignature results are obtained.

By the variation over the metric, the surface equation follows:

Kµν − K hµν = 8πGtµν. (4)

Here, the extrinsic curvature Kµν is defined to be Kµν = ∇µnν , K ≡ hµν Kµν

and the surface stress tensor tµν is tµν ≡ (2/√

h)δS1/δhµν . Eq. (4) is nothing butthe Israel junction condition [5] (see also [6]). It is customary to start from bulkAdS space:

ds2 = dz2 + l2 sinh2 z

ld�2

(4). (5)

Here �2(4) is the metric of the 4d sphere with unit radius, nz = nz = −1, nµ =

nµ = 0 (µ �= z) and ∇µnν = (1/ l) coth (z/ l) hµν . Let the surface action is

S1 =∫

d4x√

h

{− 3

8πGl+ Lm

}, (6)

where tµν = − (3/8πGl) hµν + tm µν and tm µν denotes the contribution from Lm .Assuming there is a boundary (sphere) at z = z0 Eq. (4) looks like coth (z0/ l) =1 + · · · . Here · · · expresses the possible contribution from Lm (tm muν). One candefine the brane radius R as R ≡ l sinh (z0/ l) to rewrite the brane equation in theform 0 = (1/R)

√1 + R2/ l2 − 1/ l + · · · .

As S1 one may consider more complicated action

S1 =∫

d4x√

h

{− 3

8πGl ′+ α

8πGR(4) + · · ·

}. (7)

Is brane cosmology predictable? 1421

Here R(4) is 4d scalar curvature induced on the surface and the terms containingthe higher powers of the curvature invariants are not written explicitly. This kindof the surface action has been introduced in order to cancel the divergence of thebulk action (to make AdS/CFT correspondence to be well-defined). Then

tµν =(

− 3

8πGl ′+ α

8πGR(4) + · · ·

)hµν − 2α

8πGR(4) µν + · · · , (8)

and the junction condition (4) gives the brane equation

− 3

R

√1 + R2

l2= − 3

l ′+ 6α

R2+ · · · . (9)

We now expand l.h.s. in (9) with respect to l2/R2 as − (3/R)√

1 + R2/ l2 =−3/ l − 3l/2R2 + · · · . The following choice l ′ = l, α = −l/4, · · · can con-vert (9) into an identity. Conversely, if S1 is not included, it may be induced onthe boundary. The coefficients correspond to those in the surface counterterms in[7].

If one changes the surface action S1, the junction condition (4) is also changed.For example, we consider the action like

S1 = 1

16πG

∫d4x

√h f

(R(4)

). (10)

with an arbitrary function f . For the same four-dimensional sphere one gets

tµν = 1

8πGhµν

{1

2f

(12

R2

)− 3

R2f ′

(12

R2

)}, (11)

while the junction condition is

− 3

R

√1 + R2

l2= 1

2f

(12

R2

)− 3

R2f ′

(12

R2

), (12)

It depends from the arbitrary function f . The situation is strange because the branecosmological equation could be changed rather arbitrary by the choice of f . Thequestion about brane universe predictability appears! Indeed, imagine we define afunction g(R(4)) as

g(R(4)) ≡ 1

2f

(12

R2

)− 3

R2f ′

(12

R2

)= 1

2f(R(4)

) − R(4)

4f ′(R(4)

), (13)

then f (R(4)) = −4R2(4)

∫ R(4) dx g(x)

x3 . Thus, an arbitrary junction condition in a

form as −(3/R)√

1 + R2/ l2 = g(R(4)) could be realized simply by the choice


f (R(4)) = −4R2(4)

∫ R(4) dxg(x)/x3. For example, with the choice

f(R(4)

) = 12R2(4)

∫ R(4) dx

x3

√x

12+ 1

l2, (14)

which leads to g(R(4)) = −3√

R(4)/12 + 1/ l2 = −(3/R)√

1 + R2/ l2,the junction condition (12) is identically satisfied. On the other hand, since−(3/R)

√1 + R2/ l2 < −3/ l, if g(R(4)) > −3/ l, the brane Eq. (12) has no so-

lution. This shows that brane-world is lacking the physical principle to predict inunique way the surface term and, hence, the emerging brane cosmology. One mayprescribe the AdS/CFT correspondence to define the boundary action [7] wherecurvature surface terms correspond to usual counterterms in dual QFT. Of course,this may be satisfactory only in case the string theory is the fundamental theory(which is not yet clear). As surface action is defined there, even if brane equa-tion has no solution one may take into account the other effects (brane conformalanomaly) to get the inflationary brane cosmology induced by quantum effects asis done in [8]. Note also that for the original Randall-Sundrum model [1] the sit-uation is simpler. Indeed, branes are flat there and only constant term enters thesurface action which is ambiguous only due to possible rescaling of the branetension.

Note that bulk spacetime (5) is AdS space which is not singular. As the branewhich we are considering is the sphere, or deSitter space with Euclidean signature,there is no singularity on the brane, either. Then what we have shown is that evenif there is no singularity in the bulk or on the brane, there are ambiguities in thejunction condition (4).

As in [7], one can require the surface counterterms to cancel the volume di-vergence of the bulk action. Thus, some coefficients in S1 (7) can be determined.Unfortunately, such procedure is not universal: for new bulk space one needs toredefine these coefficients again. For example, with dS/CFT correspondence (dSbulk) instead of AdS/CFT one, the coefficients are changed. Furthermore even in(5), the coefficients of the terms including the higher powers of the scalar curva-ture like R2

(4), R3(5), · · · cannot be determined only from the requirement of the

finiteness of the action. For the Einstein-Gauss-Bonnet case, the number of ambi-guities grows even more.

Another theory which received a lot of attention from brane-world point ofview is 5d EGB gravity. It may look even more attractive than Einstein gravitybecause its field equations are also of the second order but one more parameter(GB coupling constant) presents here. Various aspects of EGB gravity from branecosmology to AdS black holes and holography were investigated in refs. [9–11](see also references therein). Let us demonstrate that the same ambiguity of junc-tion condition occurs in EGB theory. First, we consider the junction condition forthe higher derivative (HD) gravity using surface counterterms found in [12,9]. Theaction of 5d R2-gravity is:

SR2 =∫

d5x√−g

{a R2

(5) + bR(5) µν Rµν

(5)

+ cR(5) µνξσ Rµνξσ

(5) + 1

κ2R(5) −

}. (15)


By introducing auxiliary fields A, Bµν , and Cµνρσ , one can rewrite the action (15)in the following form:

S =∫

d5x√−g

{a(2AR(5) − A2) + b

(2Bµν Rµν

(5) − Bµν Bµν)

+ c(2Cµνξσ Rµνξσ

(5) − Cµνξσ Cµνξσ) + 1

κ2R(5) −

}. (16)

Here κ2 = 16πG. Using the equation of the motion

A = R(5) , Bµν = R(5) µν , Cµνρσ = R(5) µνρσ , (17)

the action (16) is equivalent to (15). Let us impose a Dirichlet type bound-ary condition, which is consistent with (17), A = R(5)

∣∣on the boundary, Bµν =

R(5) µν

∣∣on the boundary, and Cµνρσ = R(5) µνρσ

∣∣on the boundary and δA = δBµν =

δCµνρσ = 0 on the boundary. However, the conditions for Bµν and Cµνρσ are, ingeneral, inconsistent. For example, even if δBµν = 0, δB ν

µ = δgνρ Bµρ �= 0. Thenone can impose boundary conditions on the scalar quantities:

A = B µµ = Cµ ν

µ ν = R(5), nµnν Bµν = nµnνCρµρν = nµnν R(5) µν. (18)

and

δA = δ(B µ

µ

) = δ(Cµ ν

µ ν

) = δ(nµnν Bµν) = δ(nµnνCρ

µρν

) = 0. (19)

We now add to the action the surface terms Sb corresponding to the Gibbons-Hawking term and S1

Sb =∫

d4x√−g

[4a∇µnµ A + 2b(nµnν∇σ nσ

+ ∇µnν)Bµν + 8cnµnν∇τ nσ Cµτνσ + 2

κ2∇µnµ

].

The above boundary action makes the variational procedure of HD theory tobe well-defined [12]. The Gauss-Bonnet combination for R2 terms correspondsto a = c, b = −4c. It is assumed the warped bulk metric is ds2 = dz2 +e2 A(z)gi j dxi dx j and there is a boundary surface (brane) at z = z0. Then if tµν

is proportional to hµν as tµν = thµν , the (function dependent) junction conditionis given by

0 = 8c(2R(5),z − 4R(5),zz,z − Ri

(5) i ,z + Ciziz ,z

)

+{

8c(6A − 24Bzz + 7R(5) zz − Ri

(5) i

) + 24

κ2

}A,z − 4t. (20)

For GB theory, the length parameter l of the bulk AdS space is given by 1/ l2 =(1/4cκ2)(1 ± √

1 + 2cκ4/3). Then Eq. (20) reduces to

0 = 6

(−12c

l2+ 1

κ2

)A,z + t . (21)


For the choice of bulk metric as in (5), Eq. (21) becomes:

−(

−12cκ2

l2+ 1

)3

R

√1 + R2

l2= 2

κ2t. (22)

Especially with the choice of S1 (10) we have

−(

−12cκ2

l2+ 1

)3

R

√1 + R2

l2= 1

2f

(12

R2

)− 3

R2f ′

(12

R2

). (23)

Hence, the junction condition could be modified rather arbitrary by the choice off . The EGB brane cosmology is not predictable as well as in Einstein theory. In[12,9], f (R(4)) has been chosen to be a constant:

f(R(4)

) = 6η. (24)

Then the junction condition (23) which defines the creation of dS universe has thefollowing form:

−(

−12cκ2

l2+ 1

)1

R

√1 + R2

l2= η, (25)

which has a solution if η < −1 + 12cκ2/ l2. The above choice of surface term(junction condition) was motivated by AdS/CFT correspondence and well-definedHD variational procedure [12]. Using it, the creation of dS and AdS brane uni-verses in EGB gravity has been carefully studied in [12] (also with the accountof brane conformal anomaly [8]). Nevertheless, misunderstanding of ambiguityof brane cosmology may lead to number of controversial claims. For instance,in [13], the same problem of creation of dS and AdS branes in 5d EGB gravity asin [12] was re-addressed. In the notations of this paper, their junction condition is

1

R

√1 + R2

l2= lλ

2

[3 − 4cκ2

l2+

(1 + 4cκ2

l2

)l2

R2

]−1

. (26)

Here λ is brane tension. Since R(4) = 12/R2, comparing (26) with (23), onetrivially finds that (26) can be reproduced from (23) with the following choice forf (R(0))

f (R(0)) = −8ζ

ξ+ 4ζ

ξ2R(0) + 4ζ

ξ3R2

(0) lnR(0)

ξ + R(0)

. (27)

Here ζ ≡ (2λ/ l)(1 + 4cκ2/ l2)−1(1 − 12cκ2/ l2)−1 andξ ≡ t (12/ l2)(1+4cκ2/ l2)−1(3−4cκ2/ l2). Of course, one can suggest many

more choices for f and define “new” cosmologies with some of them being quiterealistic.

To conclude, we demonstrated that brane cosmology is ambiguous due to thefunction dependence of junction condition (or due to the freedom in the choice ofboundary action). This is general property of any (Einstein, EGB, etc.) brane-world gravity. There should exist fundamental physical principle (subject that


brane approach is realistic) to fix this ambiguity and to make the brane cosmol-ogy to be predictable. So far, having in mind string/M-theory as a candidate forunified model it looks that only AdS/CFT related considerations may help for thispurpose (but only partially as is seen in EGB theory). It is remarkable that thispoint may be supported by using fundamental conformal symmetry (also basedon AdS/CFT) to fix the form of brane-world effective action [14].

Acknowledgements This research has been supported in part by the Monbusho of Japan undergrant n.13135208 (S.N.), by grant 2003/09935-0 of FAPESP, Brazil (S.D.O.) and by projectBFM2003-00620, Spain (S.D.O.).

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Is brane cosmology predictable?