# Is brane cosmology predictable?

###### Abstract

The creation of the inflationary brane universe in 5d bulk Einstein and Einstein-Gauss-Bonnet gravity is considered. We demonstrate that emerging universe is ambigious due to arbitrary function dependence of the junction conditions (or freedom in the choice of boundary terms). We argue that some fundamental physical principle (which may be related with AdS/CFT correspondence) is necessary in order to fix the 4d geometry in unique way.

###### pacs:

98.80.-k,04.50.+h,11.10.Kk,11.10.WxDespite the number of the efforts the construction of the theory of all fundamental interactions is still very far from the end. At the moment (super)string/M-theory remains to be the most promising candidate of unified theory. String/M-theory lives in higher dimensional space from which the early 4d universe naturally emerges in brane-world approach RS1 . Brane cosmology (for recent review and list of references, see maartens ; lidsey ) is somehow different from usual 4d cosmology due to the presence of extra terms having the higher dimensional origin. Nevertheless, there appeared the arguments that early brane universe may be quite a realistic possibility.

It remains not completely clear for all followers of brane-worlds that brane cosmology is not predictable without some additional physical principle. The junction conditions (or cosmological equations) are ambigious (up to the arbitrary function) and do not define the four-dimensional geometry in the unique way. In the present letter we demonstrate this explicitly for two models: 5d Einstein and 5d Einstein-Gauss-Bonnet (EGB) gravity. As a brane universe the inflationary (deSitter) space is considered. It is shown that creation of dS instanton always occurs just by corresponding choice of junction conditions (or by the choice of surface terms). The predictibility of brane cosmology may be achieved only when fundamental physical principle (for instance, AdS/CFT correspondence) is applied to fix 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 is the sum of the Einstein-Hilbert action , the Gibbons-Hawking surface term GH and the surface action :

(1) | |||||

(2) | |||||

(3) |

Here is the unit vector perpendicular to the surface, with direction of to be inside the bulk. The induced metric is denoted as . The Euclidean signature is used but changing and the Lorentzian signature results are obtained.

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

(4) |

Here, the extrinsic curvature is defined to be , and the surface stress tensor is . Eq.(4) is nothing but the Israel junction condition Israel (see also CR ). It is customary to start from bulk AdS space:

(5) |

Here is the metric of the 4d sphere with unit radius, , () and . Let the surface action is

(6) |

where and denotes the contribution from . Assuming there is a boundary (sphere) at Eq.(4) looks like . Here expresses the possible contribution from (). One can define the brane radius as to rewrite the brane equation in the form .

As one may consider more complicated action

(7) |

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

(8) | |||||

and the junction condition (4) gives the brane equation

(9) |

We now expand l.h.s. in (9) with respect to as . The following choice , , can convert (9) into an identity. Conversely, if is not included, it may be induced on the boundary. The coefficients correspond to those in the surface counterterms in EJM .

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

(10) |

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

(11) |

while the junction condition is

(12) |

It depends from the arbitrary function . The situation is strange because the brane cosmological equation could be changed rather arbitrary by the choice of . The question about brane universe predictability appears! Indeed, imagine we define a function as

(13) | |||||

then . Thus, an arbitrary junction condition in a form as could be realized simply by the choice . For example, with the choice

(14) |

which leads to , the junction condition (12) is identically satisfied. On the other hand, since , if , the brane equation (12) has no solution. This shows that brane-world is lacking the physical principle to predict in unique way the surface term and, hence, the emerging brane cosmology. One may prescribe the AdS/CFT correspondence to define the boundary actionEJM where curvature 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 equation has no solution one may take into account the other effects (brane conformal anomaly) to get the inflationary brane cosmology induced by quantum effects as is done in zerbini . Note also that for the original Randall-Sundrum modelRS1 the situation is simpler. Indeed, branes are flat there and only constant term enters the surface action which is ambigious only due to possible rescaling of the brane tension.

Another theory which received a lot of attention from brane-world point of view is 5d EGB gravity. It may look even more attractive than Einstein gravity because its field equations are also of the second order but one more parameter (GB coupling constant) presents here. Various aspects of EGB gravity from brane cosmology to AdS black holes and holography were investigated in refs.GB ; GB1 ; GB2 (see also refs.therein). Let us demonstrate that the same ambiguity of junction condition occurs in EGB theory. First, we consider the junction condition for the higher derivative (HD) gravity using surface counterterms found in NOjhep ; GB . The action of 5d -gravity is:

(15) | |||||

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

Here . Using the equation of the motion

(17) |

the action (Is brane cosmology predictable?) is equivalent to (15). Let us impose a Dirichlet type boundary condition, which is consistent with (17), , , and and on the boundary. However, the conditions for and are, in general, inconsistent. For example, even if , . Then one can impose boundary conditions on the scalar quantities:

(18) |

and

(19) |

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

The above boundary action makes the variational procedure of HD theory to be well-definedNOjhep . The Gauss-Bonnet combination for terms corresponds to , . It is assumed the warped bulk metric is and there is a boundary surface (brane) at . Then if is proportional to as , the (function dependent) junction condition is given by

(20) |

For GB theory, the length parameter of the bulk AdS space is given by . Then Eq.(Is brane cosmology predictable?) reduces to

(21) |

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

(22) |

Especially with the choice of (10) we have

(23) |

Hence, the junction condition could be modified rather arbitrary by the choice of . The EGB brane cosmology is not predictable as well as in Einstein theory. In NOjhep ; GB , has been chosen to be a constant:

(24) |

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

(25) |

which has a solution if . The above choice of surface term (junction condition) was motivated by AdS/CFT correspondence and well-defined HD variational procedureNOjhep . Using it, the creation of dS and AdS brane universes in EGB gravity has been carefully studied in ref.NOjhep (also with the account of brane conformal anomalyzerbini ). Nevertheless, misunderstanding of ambiguity of brane cosmology may lead to number of controversial claims. For instance, in ref.AM , the same problem of creation of dS and AdS branes in 5d EGB gravity as in ref.NOjhep was re-addressed. In the notations of this paper, their junction condition is

(26) |

Here is brane tension. Since , comparing (26) with (23), one trivially finds that (26) can be reproduced from (23) with the following choice for

(27) |

Here and . Of course, one can suggest many more choices for and define “new” cosmologies with some of them being quite realistic.

To conclude, we demonstrated that brane cosmology is ambigious due to the function dependence of junction condition (or due to the freedom in the choice of boundary 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 cosmology to be predictable. So far, having in mind string/M-theory as a candidate for unified model it looks that only AdS/CFT related considerations may help for this purpose (but only partially as is seen in EGB theory). It is remarkable that this point may be supported by using fundamental conformal symmetry (also based on AdS/CFT) to fix the form of brane-world effective actionconf .

Acknowledgments This research has been supported in part by the Monbusho of Japan under grant n.13135208 (S.N.), by grant 2003/09935-0 of FAPESP, Brazil (S.D.O.) and by project BFM2003-00620, Spain (S.D.O.).

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