Low energy branes, effective theory, and cosmology

PHYSICAL REVIEW D, VOLUME 70, 064021

Low energy branes, effective theory, and cosmology

Gonzalo A. Palma and Anne-Christine DavisDepartment of Applied Mathematics and Theoretical Physics, Center for Mathematical Sciences,

University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom(Received 10 June 2004; published 15 September 2004)

0556-2821=20

The low energy regime of cosmological BPS-brane configurations with a bulk scalar field is studied.We construct a systematic method to obtain five-dimensional solutions to the full system of equationsgoverning the geometry and dynamics of the bulk. This is done for an arbitrary bulk scalar fieldpotential and taking into account the presence of matter on the branes. The method, valid in the lowenergy regime, is a linear expansion of the system about the static vacuum solution. Additionally, wedevelop a four-dimensional effective theory describing the evolution of the system. At the lowest orderin the expansion, the effective theory is a biscalar tensor theory of gravity. One of the main features ofthis theory is that the scalar fields can be stabilized naturally without the introduction of additionalmechanisms, allowing satisfactory agreement between the model and current observational constraints.The special case of the Randall-Sundrum model is discussed.

DOI: 10.1103/PhysRevD.70.064021 PACS numbers: 04.50.+h

I. INTRODUCTION

Brane-world models have become an important subjectof research in recent years. The simple and intuitivenotion that the standard model of physics is embeddedin a hypersurface (brane), located in a higher dimen-sional space (bulk), has significant implications for grav-ity, high energy physics and cosmology. It is stronglymotivated by recent developments in string theory andits extended version, M theory [1–3]. Additionally, it hasbeen proposed on a phenomenological basis [4–6] toaddress many questions which cannot be answered withinthe context of the conventional standard model of phys-ics, such as the hierarchy problem and the cosmologicalconstant problem [7–12].

The basic idea is that gravity—and possibly otherforces—propagate freely in the bulk, while the standardmodel’s fields are confined to a four-dimensional brane(see [13–16] for comprehensive reviews on brane-worldmodels). In this type of model, constraints on the size ofextra dimensions are much weaker than in Kaluza-Kleintheories [17], although, Newton’s law of gravity is stillsensitive to the presence of extra dimensions. Gravity hasbeen tested only to a tenth of millimeter [18], therefore,possible deviations to Newton’s law, below that scale, canbe envisaged. In general, one of the main modifications togeneral relativity offered by brane models, is the appear-ance, in the effective Einstein’s equations of the system,of novel terms that depend quadratically on the energy-momentum tensor of the matter content of the observedUniverse. When these terms can be neglected the branesystem is in the low energy regime. It is well known thatthe most recent cosmological era must be consideredwithin the low energy regime. Despite this fact, thereare still important modifications to general relativitythat must be considered, and current observational con-

04=70(6)=064021(20)$22.50 70 0640

straints are highly relevant for the phenomenology ofbrane models.

Cosmologically speaking, the evolution of the branesin the bulk is directly related to the evolution of theobserved Universe [19,20]. So far, most of the cosmologi-cal considerations of brane worlds have been centered onthe models proposed by Randall and Sundrum [9,10].They considered warped geometries in which the bulkspace is a slice of anti-de Sitter space-time. One of themain problems of these scenarios is the stabilization ofnew extra degrees of freedom appearing from the con-struction. More precisely, when the four-dimensionalequations of motion for the branes are considered, anew scalar degree of freedom, the radion, emerges. Thestabilization of the radion is an important phenomeno-logical problem, and many different mechanisms to sta-bilize it have been proposed [21,22]. Clearly, branemodels which only consider the presence of gravity inthe bulk, such as the Randall-Sundrum model, are notfully satisfactory from a theoretical point of view. Thereis no reason a priori for gravity to be the only forcepropagating in the bulk, and other unmeasured forcesmay also be present. For instance, string theory predictsthe existence of the dilaton field which acts as a masslesspartner of the graviton. If such forces are present in thebulk, they should be suppressed by some stabilizationmechanism in order to evade observation (similar to thecase of the radion field in the Randall-Sundrum model).The problem of finding a stabilization mechanism is anactive field of research.

In this paper we are going to consider a fairly generalbrane-world model (BPS branes), which has been moti-vated as a supersymmetric extension of the Randall-Sundrum model [23–27] (see [28,29] for phenomenologi-cal motivations). It consists of a five-dimensional bulkspace with a scalar field , bounded by two branes,�1 and

21-1 2004 The American Physical Society

GONZALO A. PALMA AND ANNE-CHRISTINE DAVIS PHYSICAL REVIEW D 70 064021

�2. The main property of this system is a special bound-ary condition that holds between the branes and the bulkfields, which allows the branes to be located anywhere inthe background without obstruction. In particular, a spe-cial relation exists between the scalar field potentialU� �,defined in the bulk, and the brane tensions UB� 1� andUB�

2� defined at the position of the branes. This is theBPS condition. Because of the complexity of this setup,the main efforts of previous works have been focused onthe special case of dilatonic branes, UB� � / e

� , where� is a parameter of the theory [30,31]. This particularpotential is well motivated by supergravity in singularspaces and string theory. For instance, when � � 0, theRandall-Sundrum model is retrieved, meanwhile, forlarger values, � � O�1�, one obtains the low energy ef-fective action of heterotic M theory, taking into accountthe volume of the Calabi-Yau manifold [32].Nevertheless, other classes of potentials also have physi-cal motivation and should be considered. For instance, infive-dimensional supergravity, one can expect the linearcombinationUB� � � V1e�1 � V2e�2 , where �1 and �2

are given numbers [25]. It must be stated, however, that upto now the problem of assuming an arbitrary potentialUB� � has not been considered in detail, and therefore,the low energy regime of this type of system is not wellunderstood. For example, it is not known how the gravi-tational fields nor the bulk scalar field behave near thebranes, and the general belief is that these issues should beinvestigated via numerical techniques.

Here we study the low energy regime of brane modelswith a bulk scalar field. One of the most important as-pects is that we shall not restrict our treatment to anyparticular choice of the BPS potential, allowing a clearerunderstanding of five-dimensional BPS systems, and fo-cus our study on the problems mentioned previously. Inparticular we shall develop a systematic method to obtainfive-dimensional solutions for the gravitational fields andthe bulk scalar fields. Additionally, by adopting the pro-jective approach, we develop an effective four-dimensional theory, valid in the low energy regime, andindicate that it is equivalent to the moduli-space approxi-mation. This effective theory is a biscalar tensor theory ofgravity of the form:

S �1

k 25

Zd4x

��g

p�R�

3

4g��ab@�

a@� b �

3

4V�

�S1��1; A21g�� S2��2; A

22g��; (1)

where a � 1; 2 labels the branes, a corresponds to theprojected bulk scalar field at the brane �a, �ab is a�-model metric of the biscalar theory, S1 and S2 are theactions for matter fields �1 and �2 living in the respec-tive branes, and A1 and A2 are warp factors dependent onthe scalar fields. Obtaining this effective theory is a greatachievement that allows the study of this class of models

064021

within the approach and usual techniques of multiscalartensor theories [33,34].

This paper is organized as follows: In Sec. II, wereview brane systems with a bulk scalar field. There, weillustrate in a more technical way some of the problemsthat this paper addresses. In Sec. III, we work out andanalyze the static solutions for the vacuum configurationof BPS branes. This enables us to construct the low energyregime expansion in Sec. IV. Then, in Sec. V, we developthe four-dimensional effective theory valid in the lowenergy regime. There, we will focus our attention on thezeroth order effective theory which is a biscalar tensortheory of gravity. In Sec. VI, we study the cosmology ofthe effective theory. For instance, we consider the con-straints coming from current observations on deviationsto general relativity and analyze the stabilization problemof BPS systems. It will be found that, in order to stabilizethis type of system it is enough to have the appropriatepotential UB, and therefore no exotic mechanisms arerequired in general. The conclusions are summarized inSec. VII.

II. BPS BRANES WITH A BULK SCALAR FIELD

We start this section with the introduction of five-dimensional BPS configurations, deriving the five-dimensional equations of motion governing the dynamicsof the system and deducing the projected equations at theposition of the branes. These equations depend on theconfiguration of the bulk as well as on the matter contentin the branes; we analyze the difficulties arising from thisdependence to find solutions to realistic cosmologicalconfigurations.

A. Basic configuration

Let us consider a five-dimensional manifold M pro-vided with a coordinate system XA, with A � 0; . . . ; 4. Weshall assume the special topology M � � S1=Z2,where � is a fixed four-dimensional Lorentzian manifoldwithout boundaries and S1=Z2 is the orbifold constructedfrom the one-dimensional circle with points identifiedthrough a Z2 symmetry. The manifold M is bounded bytwo branes located at the fixed points of S1=Z2. Let usdenote the brane surfaces by �1 and �2, respectively. Weshall usually refer to the space bounded by the branes asthe bulk space. We will consider the existence of a bulkscalar field living in M with boundary values, 1 and 2, at the branes. Consider also the presence of a bulkpotential U� � and brane tensions V 1� 1� and V 2� 2�(which are potentials for the boundary values 1 and 2).Additionally, we will consider the existence of matterfields �1 and �2 confined to the branes. Figure 1 showsa schematic representation of the present configuration.

Given the present topology, it is appropriate to intro-duce foliations with a coordinate system x� describing �(as well as the branes �1 and �2) where � � 0; . . . ; 3.

-2

FIG. 1. Schematic representation of the five-dimensionalbrane configuration. In the bulk there is a scalar field witha bulk potential U� �. Additionally, the bulk space is boundedby branes, �1 and �2, with tensions given by V 1� 1� andV 2�

2� respectively, where 1 and 2 are the boundary valuesof .

LOW ENERGY BRANES, EFFECTIVE THEORY, AND . . . PHYSICAL REVIEW D 70 064021

Additionally, we can introduce a coordinate z describingthe S1=Z2 orbifold and parametrizing the foliations. Withthis decomposition the following form of the line elementcan be used to describe M:

ds2 � �N2 �N�N��dz2 � 2N�dx�dz�g��dx�dx�: (2)

Here N and N� are the lapse and shift functions for theextra-dimensional coordinate z and, therefore, they canbe defined up to a gauge choice (for example, theGaussian normal coordinate system is such that N� �0). Additionally, g�� is the pullback of the induced metricon the four-dimensional foliations, with the ��;�;�;��signature. The branes are located at the fixed points of theS1=Z2 orbifold, denoted by z � z1 and z � z2, where 1and 2 are the labels for the first and second branes, �1 and�2. Without loss of generality, we shall take z1 < z2.

The total action of the system is

Stot � SG � Sbulk � SBR; (3)

where SG is the action describing the pure gravitationalpart and is given by SG � SEH � SGH, with SEH theEinstein-Hilbert action and SGH the Gibbons-Hawkingboundary terms; Sbulk is the action for the matter fieldsliving in the bulk, which can be decomposed into Sbulk �S � Sm. Here S is the action for the bulk scalar field (including its boundary terms), while Sm is the action forother matter fields (in the present work we will notspecify this part of the theory). Finally SBR is the actionfor the matter fields, �1 and �2, confined to the branes.

064021

Let us see in more detail the different contributionsappearing in Eq. (3). First, note that with the presentparametrization for M, we can write an infinitesimalproper element of volume as dV � dzd4x

��g

pN. Then,

it is possible to express SG in the following way:

SG � SEH � SGH

�1

2 25

ZS1=Z2

dzZ�d4x

��g

pN�R� �K��K

�� K2�;

(4)

where R is the four-dimensional Ricci scalar constructedfrom g��, and 2

5 � 8#G5, with G5 the five-dimensionalNewton’s constant. Additionally, K�� g0�� r�N� �r�N�=2N is the extrinsic curvature of the foliations andK its trace (the prime denotes derivatives in terms of z,that is 0 � @z, and covariant derivatives, r�, are con-structed from the induced metric g�� in the standardway). On the other hand the action for the bulk scalarfield, S , can be written in the form

S � �3

8 25

Zdzd4x

��g

pN�� 0

N

�2� �@ �2

� 2N�

N2 @� @z �U� �� S1 � S2 ; (5)

where S1 and S2 are boundary terms given by

S1 � �3

2 25

Z�1

d4x��g

pV 1�

1�; (6)

S2 � �3

2 25

Z�2

d4x��g

pV 2�

2�; (7)

at the respective positions, z1 and z2. In the above ex-pressions U� � is the bulk scalar field potential, whileV 1� 1� and V 2� 2� are boundary potentials defined atthe positions of the branes (also referred to as the branetensions). In the present case (BPS configurations), weshall consider the following general form for the poten-tials:

U � U� u; (8)

V 1 � UB � v1; (9)

V 2 � UB � v2; (10)

where U and UB are the bulk and brane superpotentials,and the potentials u, v1 and v2 are such that juj � jUjand jv1j, jv2j � jUBj. In this way, the system is domi-nated by the superpotentials U and UB. The most impor-tant characteristic of this class of system is the relationbetween U and UB (the BPS relation), given by

U � �@ UB�2 �U2

B: (11)

As we shall see later in more detail, the BPS property

-3


mentioned earlier (the fact that the branes can be locatedanywhere in the background, without obstruction) comesfrom the relation above. This specific configuration, whenthe potentials u, v1, v2 � 0 and no fields other than thebulk scalar field are present, is referred to as the BPSconfiguration. It emerges, for instance, in models likeN � 2 supergravity with vector multiplets in the bulk[25]. More generally, it can be regarded as a supersym-metric extension of the Randall-Sundrum model. In par-ticular when UB is the constant potential, the Randall-Sundrum model is recovered with a bulk cosmologicalconstant �5 � �3=8�U � ��3=8�U2

B, and the usual fine-tuning condition is given by Eq. (11). The presence of thepotentials u, v1 and v2 are generally expected fromsupersymmetry breaking effects. However, their specificform is not known and, up to now, they must be arbitrarilypostulated [35].

Finally, for the matter fields confined to the branes, weshall consider the standard action:

SBR � S1��1; g��z1� � S2��2; g��z2�; (12)

where �1 and �2 denote the respective matter fields, andg��za� is the induced metric at position za. In this articlewe shall assume that the matter fields are minimallycoupled to the bulk scalar field. That is to say, we willassume that SBR does not depend explicitly on .Nevertheless, it is important to point out that, in general,one should expect some nontrivial coupling between thebulk scalar field and the matter fields �1 and �2, thatcould lead to the variation of constants in the brane-worldscenario [36]. Despite these comments, we show that, in aminimally coupled system, an effective coupling is gen-erated between the matter fields and the bulk scalar field.

B. Equations of motion

In this subsection we derive the equations of motiongoverning the dynamics of the fields living in the bulk.These equations are obtained by varying the action Stotwith respect to the gravitational and bulk scalar fields,without taking into account the boundary terms. Thesewill be considered in the next subsection, in order todeduce a set of boundary conditions for the bulk fields.

The five-dimensional Einstein’s equations of the sys-tem can be derived through the variation of the totalaction Stot with respect to N, N� and g��. Respectively,these equations are found to be

R� �K��K�� K2 � 2 25 ~S; (13)

r��K�� K'�� 25 ~J�; (14)

G�� E�� 12g��K��K

�� K2 � KK�� K��K��

�23

25�~T��

14g��

~T�� 3~S�: (15)

The above set of equations deserves a detailed descrip-

064021

tion. G��, in (15), is the Einstein tensor constructed fromthe induced metric g��. The quantities ~T��, ~J�, and ~S arethe four-dimensional stress energy-momentum tensor,vector and scalar, respectively, characterizing the mattercontent in the bulk space. They are defined as

~S � �1��g

p'Sbulk'N

; ~J� � �1��g

p'Sbulk'N� ;

~T�� 2

N��g

p'Sbulk'g��

; (16)

where Sbulk � S � Sm is the action for the matter fieldsliving in the bulk. The above quantities are decomposi-tions of the usual five-dimensional stress energy-momentum tensor, TAB. To be more specific, the defini-tions in (16) satisfy ~T�� XA;�XB;�TAB, ~J� � �XA;�nBTAB,and ~S � �nAnBTAB, where XA;� is the pullback betweenthe coordinate system x� of the foliations and the five-dimensional coordinate system XA of M, and where nA isthe unit-vector field normal to the foliations. In particu-lar, the contributions coming from the scalar field action,S , are given by

~S� � � �3

8 25

�1

N2 � 0�2 � 2

N�

N2 @� @z � �@ �2 �U

�;

~J� �� 3

4 25

1

N@z @� ;

~T� ��

3

4 25

�1

2g��

�1

N2 � 0�2 � 2

N�

N2 @� @z

��@ �2 �U

�� @� @�

�: (17)

Also in (15) is E��, the projection of the five-dimensionalWeyl tensor, CCADB, on the foliations given by E�� XA;�XB;�nCnDC

CADB. In the present setup, this can be written

as

E�� K��K��

1

N�r�r�N �LNnK��

�1

3 25

�~T��

1

2g�� ~T

�� ~S�

�; (18)

where LNn is the Lie derivative along the vector fieldNnA. At this point it is convenient to adopt the gaugechoice N� � 0, which corresponds to the choice ofGaussian normal coordinates. With this gauge, one hasLNn � @z, and the treatment of the entire system isgreatly simplified. Next, we can obtain the equation ofmotion for the bulk scalar field . Varying the action withrespect to gives

g��r��N@� � � � 0=N�0 � K 0 �N2

@U@

: (19)

(Recall that we are now using N� � 0.)

-4


It is possible to see that Eqs. (13) and (15) are thedynamical equations for the gravitational sector (noticethat second order derivatives in terms of z are only presentin the projected Weyl tensor E��). Meanwhile, Eq. (14)constitutes a restriction for the extrinsic curvature K�� interms of the matter content of the bulk. As we shall soonsee in more detail, the conservation of energy momentumon the branes is a direct consequence of Eq. (14).

C. Boundary conditions

We can now turn to the boundaries of the system. Theformer variations leading to Eqs. (13)–(15) and (19) alsogive rise to boundary conditions that must be respected bythe bulk fields at positions z1 and z2. They emerge whenwe take into account the boundary terms present in theaction (including the matter fields in the branes). For thefirst brane, �1, these conditions are found to be

fK�� g��Kgjz1 �32�UB � v1g�� 25T

1��; (20)

12 f

0=Ngjz1 � @ �UB � v1; (21)

while for the second brane, �2, these conditions are

fK�� g��Kgjz2 � �32�UB � v2g�� 25T

2��; (22)

12 f

0=Ngjz2 � �@ �UB � v2: (23)

Equations (20) and (22) are the Israel matching conditionsand Eqs. (21) and (23) are the BPS matching conditionsfor the bulk scalar field. In the previous expressions thebrackets denote the difference between the evaluation ofquantities at both sides of the branes, i.e., ffgjz �lim/!0�f�z� /� � f�z� /�. Additionally, Ta�� denotesthe energy-momentum tensors of the matter content inthe brane �a, and is defined by the standard expression:

Ta�� 2��g

p'Sa'g��

za

: (24)

Since we are considering an orbifold with a Z2 symmetry,and the branes are positioned at the fixed points, theconditions (20)–(23) can be rewritten as

K�� g��K �3

4�UB � v1g��

252T1��; (25)

0 � N@ �UB � v1; (26)

for the first brane, �1, and

K�� g��K �3

4�UB � v2g��

252T2��; (27)

0 � N@ �UB � v2; (28)

for the second brane, �2. These conditions are of utmostimportance. They relate the geometry of the bulk with the

064021

matter content in the branes. In the next subsection weshall examine in more detail how these conditions arerelated to the four-dimensional effective equations in thebranes and analyze some of the open questions.

D. Short analysis and open questions

Equations (13)–(15) and (19) can be evaluated at theposition of the branes, with the help of the boundaryconditions (25)–(28), to obtain a set of four-dimensionaleffective equations. This is the projective approach. Forinstance, evaluating Eqs. (13) and (15) at the position ofthe first brane, we find

R � � 254V 1T � g��

3

4W �

3

4�@ �2 � g��m�

�� ;

(29)

G�� 25

4V 1T�� m�

�� E��

�1

2

�@� @� �

5

8g��@ �2

��

3

16g��W; (30)

where ��m�� , �� and W have been defined as

��m�� 2

3 25�~T�m�� 1

4g��~T�m�� 3~S�m��; (31)

�� 45

4

�1

3TT�� T��T

��

1

2g��T��T

��

�1

6g��T

2

�; (32)

W � U�

�@V 1

@

�2�V 2

1: (33)

Notice that we have dropped the index labeling the firstbrane in the energy-momentum tensor (that is, we havetaken T�� T1

�� and T � g��T1��). Similarly, the pro-

jection of Eq. (19) on the first brane leads us to the four-dimensional effective bulk scalar field equation. This isfound to be

� � �1

N@�N@� �

256

@V 1

@ T ��

1

2

@W@

�@V 1

@ @2v1@ 2 ; (34)

where we have defined the loss parameter � of thesystem as

� �1

N� 0=N � @ UB

0: (35)

Additionally, from the evaluation of Eq. (14) at positionz1, one finds the energy-momentum conservation relationfor the matter fields of the first brane:

r�T�� 2~J�m�

� : (36)

-5


Note that if u � v1 � 0 then W � 0. The precise form of~T�m�� , ~J�m�

� and ~S�m�, in Eqs. (29), (30), and (36), depend onthe bulk matter fields considered in Sm. From now on wewill not consider any contribution from Sm and thereforewe shall take ~T�m�

�� ~J�m�� ~S�m� � 0 (that is, the only

fields living in the bulk are the gravitational fields g��and N, and the bulk scalar field ).

Equations (29), (30), and (34), are the projected equa-tions describing the theory in the first brane (an analog setof equations can be obtained for the second brane). Tosolve them, it is important to know the precise form of theprojected Weyl tensor E�� and the loss parameter � . Inorder to find a complete solution of the entire system it isnecessary to find a solution for these quantities. Theycontain the necessary information from the bulk, orequivalently, from a brane frame point of view; theypropagate the information from one brane to the other.A consistent method to compute E�� and � is an openquestion.

Observe that the four-dimensional effective Newton’sconstant, G4, can be identified as 8#G4 � 25V 1=4 and,therefore, it is a function of the projected bulk scalar field . Additionally, it can be seen that there are two scalardegrees of freedom. Naturally, one of them is the pro-jected bulk scalar field . The second one corresponds tothe projected value of lapse functionN, which is normallyreferred to as the radion field. In the absence of super-symmetry breaking terms u, v1 and v2, these scalardegrees of freedom are massless and are the moduli fields.The stabilization of moduli fields is one of the mostimportant phenomenological issues in brane models toensure agreement with observations.

The term ��, in Eqs. (30) and (34), contains qua-dratic contributions from the energy-momentum tensorT��. Therefore, �� is relevant at the high energy regimeand it will turn out to be important for the early universephysics. At more recent times, in order to agree withobservations, this term must be negligible, which isonly possible if 25jUBj � jTj. This is the low energyregime. Neglecting �� makes the previous equationsof motion take the form:

R � � 254UBT �

3

4�@ �2; (37)

G�� 254UBT�� E��

1

2

�@� @� �

5

8g��@ �2

�;

(38)

� � �1

N@�N@� �

256

@UB

@ T � � ; (39)

where we have, for simplicity, also dropped the termscontaining u and v1. Recall the additional equation ex-pressing energy-momentum conservation on the first

064021

brane, which now reads

r�T�� 0: (40)

This version for the equations of motion is much simplerthan the full system, however, the difficulties mentionedabove are still present and many questions remain to beanswered. For example, the usual approach is to neglectthe effects of the loss parameter � and take it to be zero.If this is the case, observe that the projected scalar field ,in Eq. (39), is driven by the derivative @ UB times thetrace of the energy-momentum tensor. One could there-fore expect the scalar field to be stabilized provided theappropriate form of UB. This possibility, however, willdepend on the dynamics of the radion field N, which, atthe same time depends on the dynamics of the interbranesystem. The stabilization of the scalar field and the radionare usually solved by the introduction of new potentialterms for at the branes and the bulk. In the present casethat could be the role of u, v1 and v2.

In the rest of the paper we shall focus our efforts tosolve these problems in the low energy regime. For in-stance, we will see that it is possible to introduce aprocedure to compute E�� and � with any desire pre-cision. Additionally, we show that it is possible to stabi-lize the scalar degrees of freedom solely with theintroduction of the appropriate potential UB (that is,without the need of supersymmetry breaking terms u,v1 and v2).

III. STATIC VACUUM CONFIGURATION

Before studying realistic solutions in detail, namely,when matter fields are present in the branes, let us analyzethe static vacuum solutions to this system. This will behighly relevant for the development of the low energyexpansion and the effective theory, to be worked out in thenext sections.

A. Static vacuum solution

Let us consider the presence of the bulk scalar field and gravitational fields N and g�� and assume that thematter fields �1 and �2 (and supersymmetry breakingterms) are absent. In this case, we are mainly concernedwith static vacuum solutions to the system of Eqs. (13)–(15) and (19). Therefore, we shall assume that , N andg�� are such that

@� � 0; @�N � 0; and G�� 0; (41)

where G�� is the Einstein’s tensor constructed from g��.To solve the system, it is sensible to consider the follow-ing form for the induced metric:

g�� !2�z�~g��x�; (42)

where ~g��x� is a metric that depends only on the space-time coordinate x, and which necessarily satisfies the

-6


four-dimensional vacuum Einstein’s equations ~G�� 0.In this way, all the dependence of the induced metric onthe extra-dimensional coordinate z is contained in thewarp factor !�z�. With this form of the metric, the set ofmatching conditions (25)–(28) can be rewritten in thefollowing simple way:

!0

!� �

1

4NUB; (43)

0 � N@UB

@ : (44)

More importantly, it is possible to show that these rela-tions solve the entire system of Eqs. (13)–(15) and (19).This important fact constitutes one of the main propertiesof BPS systems. It means that the branes can be arbi-trarily located anywhere in the background, without ob-struction. It should be clear that when matter is allowed toexist in the branes, the boundary conditions (43) and (44)will not continue being solutions to the system ofEqs. (13)–(15), and the static configurations will not bepossible in general. The introduction of matter in thebranes will drive the branes to a cosmological evolution.

In the static vacuum solution expressed throughEqs. (43) and (44), the dependence of the lapse functionN, in terms of the extra-dimensional coordinate z, iscompletely arbitrary, though it must be restricted to bepositive in the entire bulk, and its precise form willcorrespond to a gauge choice. Let us assume that �z�satisfies Eqs. (43) and (44), and that it has boundaryvalues 1 and 2 defined by

1 � �z1� and 2 � �z2�: (45)

Since we are interested in the static vacuum solution, 1

and 2 are just constants. The precise form of �z�, as afunction of z, depends on the form ofUB� � and the gaugechoice for N. However, it is not difficult to see that 1 and 2 are the only degrees of freedom necessary to specifythe BPS state of the system. That is, given a gauge choicefor N, in general we have

� �z; 1; 2� and N � N�z; 1; 2�: (46)

Observe, additionally, that !�z� can be expressed interms of �z� in a gauge independent way. Using bothrelations, (43) and (44), we find

!�z� � exp��1

4

Z �z�

1��1� �d

�; (47)

�� 1

UB

@UB

@ : (48)

In the last equations we have normalized the solution!�z�in such a way that the induced metric to the first brane is~g��, though this choice is not strictly necessary. The

064021

induced metric on the second brane is, therefore, confor-mally related to the first brane, with a warp factor !�z2�.

Let us emphasize the fact that, given a solution �z�and a gauge choice N, satisfying Eqs. (43) and (44), theentire system can be specified by providing the degrees offreedom 1, 2 and ~g��. In the next section we shall seethat when matter fields are considered, the vacuum solu-tion is perturbed and 1, 2 and ~g�� must be promoted tosatisfy nonvacuum equations of motion. The resultingtheory will be a biscalar tensor theory of gravity, withthe two scalars given by 1 and 2.

B. Bulk geometry

It is instructive to know the behavior of the scalar field and warp factor ! in the bulk. For instance, a relevantquestion is what the conditions are for a singularity to bepresent in the bulk? Let us start by noting that Eqs. (43)and (44) give us the way in which the scalar field andthe warp factor ! behave as functions of the properdistance, d2 � Ndz, in the bulk space. If UB > 0, thenthe warp factor will decrease in the z direction, while ifUB < 0 then the warp factor will be increasing. Similarly,if @ UB > 0, then the scalar field will increase in the zdirection, and if @ UB < 0, then it will decrease.Moreover, ifUB�

1�> 0 then the first brane has a positivetension while ifUB�

1�< 0 then it has a negative tension.The situation for the second brane is similar: ifUB�

2�>0 then it has a negative tension while ifUB�

2�< 0 then ithas a positive tension. In this way, it is possible to see thatin general the two branes will not necessarily have oppo-site tensions. Figure 2 sketches the different possiblebehaviors for and ! as functions of z.

From Eq. (44) we see that the infinitesimal properdistance, d2 , in the extra-dimensional direction can bewritten as

d2 �

�@UB

@

��1d : (49)

Observe from this last relation that extremum points ofthe potential UB, given by the condition @ UB � 0, canbe only reached at an infinite distance away in the bulk.This is easily seen from the fact that d2 ! �1 as@ UB ! 0. Thus @ UB cannot change sign in the bulkspace and, therefore, �z� is a monotonic function of z.This is not the case, however, for the warp factor !. Thiscan increase or decrease according to the sign of UB� �for a given value of . The fact that �z� is monotonic inthe bulk space allows the possibility of parametrizing thebulk with instead of z. This is an important result thatwill be heavily exploited in the rest of this paper.

Finally, let us examine the possibility of having singu-larities at a finite position in the bulk. From Eq. (47) wecan see that singularities will appear in the bulk when-ever !�z� � 0 or !�z� � �1. In general, these singular-ities will be located at points, z1, where the following

-7

FIG. 2. The figure shows the four different possibilities for the behavior of and ! as a function of z, according to the signs of@ UB and UB. The parameters of the figures are as follows: (a) @ UB > 0 and UB > 0 (or �> 0). (b) @ UB > 0 and UB < 0 (or�< 0). (c) @ UB < 0 and UB > 0 (or �< 0). (d) @ UB < 0 and UB < 0 (or �> 0).


integral diverges:

Z �z�

1��1� �d : (50)

Let us designate 1 � �z1� to the value of the scalarfield at which (50) becomes divergent. Then, fromEq. (49), it is possible to see that for a singularity to belocated at a finite proper distance in the bulk, the follow-ing additional condition needs to be satisfied:

Z 1

1

�@UB

@

��1d

<�1: (51)

Now, notice that the integral in (50) will diverge either if@ UB� 1� ! 0 or UB� 1� ! �1. We have alreadystudied the first case, which necessarily happens at in-finity. The second case, UB� 1� ! �1, will be associ-ated with singularities of the type !! 0. For example,the only way to approach a singularity of the typeUB� � ! �1, with z! z�1, that is from the left, willbe with @ UB > 0. This corresponds to UB� � ! �1,and therefore to a warp factor going as !! 0 (the sameargument follows for z! z�1). This means that the only

064021

type of singularity, at a finite location in the bulk, will beof the type !! 0.

Summarizing, we have seen how the bulk geometry ofthe vacuum static system is completely determined by thebehaviors of and !. is a monotonic function of z,while! can increase or decrease depending on the sign ofUB. The only singularities possible in the bulk are of theform !! 0. In general, this is a good reason to considertwo-brane models in the context of BPS systems: thesingularity can be shielded from the first brane, andmade to disappear from the bulk, with the presence ofthe second brane. However, the second brane can beattracted towards the singularity and eventually hit it.

C. A few examples

To gain some experience with BPS configurations, letus briefly study the static vacuum solution for a fewchoices of the potential UB. In particular, it will be usefulto see how the Randall-Sundrum model arises as a par-ticular case of the present type of system. Let us start byanalyzing the case of dilatonic branes, UB � V0e� ,where � is an arbitrary constant. As already said in

-8


Sec. I, this form of the potential is motivated by heteroticM theory, and is also predicted in five-dimensional su-pergravity. Let us first consider the gauge N > 0 a con-stant, and, for simplicity, choose the positions of thebranes at z1 � 0 and z2 � r, with r > 0. Then, it followsthat the solution to the system is given by

�z� � �1

�ln��Nz� s��2V0; (52)

!�z� � ��Nz� s��2V01=4�2

exp� 1=4��; (53)

where N and s can be expressed as

N � �1

r�2V0

�e�� 2� e��

1�; (54)

s � �1

�2V0

e�� 1: (55)

For the singularities, there are two generic cases. WhenV0 > 0, the first brane is a positive tension brane and thesecond one a negative tension brane, and a singularity ofthe type ! � �1 will exist at z � �1, while a singu-larity of the type ! � 0 will exist at the finite positionz1 � �s=N � re��

1=�e��

1� e��

2�> r. Thus, it is

not difficult to see that if � 2 ! �1, then the secondbrane hits the singularity ! � 0. Figure 3 shows thesolutions for �z� and !�z� in the present case (with thechoice �< 0). In the case V0 < 0, the first brane corre-sponds to a negative tension brane and the second brane toa positive tension brane, with singularities happening atthe same coordinates but with opposite signs.

The Randall-Sundrum scenario is easily obtained byletting �! 0. In this limit, the above solutions can be

FIG. 3. The figure shows the solutions of �z� and !�z� forthe exponential case, when the gauge N constant is adopted.

064021

reexpressed as

�z� � 0; (56)

!�z� � e��Nz; (57)

where � � �1=4�UB. In this case, the singularities arelocated at infinity. Note that in this case the only degree offreedom is N, the radion field.

IV. LOW ENERGY REGIME EXPANSION

We are now in a position to deduce the equations ofmotion governing the low energy regime. They consist ina linear expansion of the fields about the static vacuumsolution found in the last section. As we shall see, theseequations can be put in an integral form, allowing theconstruction of a systematic scheme to obtain solutions,order by order. The zeroth order solution of this expansioncorresponds to the static vacuum solutions with the in-tegration constants 1 and 2, and the metric ~g�� pro-moted to be space-time dependent.

A. Low energy regime equations

Let us develop the equations for the low energy regime.To start, assume that 0,!0 and N0 are functions of x andz that satisfy the BPS conditions:

!00

!0

� �1

4N0UB� 0�; (58)

00 � N0

@UB

@ 0; (59)

and let us define the bulk scalar field boundary values as 10 � 0�z1� and 2

0 � 0�z2� (note that they are func-tions of the space-time coordinate x). The solution for thewarp factor, !0, is found to be

!0�z; x� � exp��1

4

Z 0

10

��1� �d �; (60)

where the z and x dependence enter through the functions 10�x� and 0�z; x� in the integration limits. It will be

convenient to define the warp factor between the twobranes as !! � !0�z2�, or more explicitly, as

!!�x� � exp��1

4

Z 20

10

��1� �d �: (61)

Now, we would like to study the perturbed system aboutthe static vacuum solution.With this purpose, let us definethe following set of variables, h��, ’ and 7, as

g�� !20�~g�� h��; (62)

� 0 � ’; (63)

-9


N � N0e7; (64)

where g��, and N satisfy the equations of motion (13),(15), and (19), taking into account the presence of matterin the branes and supersymmetry breaking terms in thepotentials. Additionally, ~g�� depends only on the space-time coordinate x. Note that if there is no matter contentin the branes and u � v1 � v2 � 0, then we can takeh�� ’ � 7 � 0, and the fields g��, and N of theprevious definition would correspond to the static vacuumsolution discussed in the last section, with 1

0 and 20 the

two constant degrees of freedom. Therefore, the functionsh��, ’ and 7 are linear deviations from the vacuumsolution of the system. Since we are interested in thelow energy regime, we consider the case jh��j � j~g��j,j’j � j 0j and j7j � 1. Now, if we insert these previousdefinitions in the equations of motion (13), (15), and (19),and neglect second order quantities in terms of h��,’ and7 we obtain the equations of motion for the low energyregime. First, Eq. (13) leads to

N0UB� 0�~g�8h0�8 � 2N0

@UB

@ 0’0 � N2

0

�2U07�

@U0

@ 0’�

�N2

0

!20

X: (65)

Equation (15) leads to the four-dimensional Einstein’sequations:

h00�� ~g��~g�8h00�8 �

�N0UB� 0� �

@N0

@ 0

@UB

@ 0

�

�h0�� ~g��~g�8h0�8� �3

2N0UB� 0�70~g��

�3

2N0@UB

@ 0’0~g��

3

4N2

0

�2U07�

@U0

@ 0’�~g��

� 2N2

0

!20

Y��: (66)

And finally, Eq. (19) [with the help of Eq. (13)] leads to

’00 �

�N0UB� 0� �

@N0

@ 0

@UB

@ 0

�’0 � N0

@UB

@ 070

�1

2N0@UB

@ 0~g��h0��

N20

2

�27

@U0

@ 0�@2U0

@ 20

’�

�N2

0

!20

�Z�

1

2�0X

�: (67)

In the previous equations, U0 � U� 0� and �0 � �� 0� .Equations (65) and (66) correspond to the linearizedEinstein’s equations, while Eq. (67) corresponds to thelinearized bulk scalar field equation. In the previous set ofequations we have defined, for the sake of notation, thefunctions X, Y�� and Z, in the following way:

064021

X � !20

��@ �2 �

4

3R� u

�; (68)

Y �� G�� 1

N�g��N �r�r�N� �

3

4

�1

2g��@ �

2 � @� @� �1

2g��u

�; (69)

Z � !20

�1

2

@u@ 0

�1

Ng��r��N@� �

�: (70)

Note that in the functions X, Y�� and Z, the quantitiesg��, and N are not explicitly expanded. Their expan-sions can be considered in the following way:

X �X0�X; Y��Y0��Y��; Z�Z0�Z; (71)

where X0, Y0�� and Z0 are the zeroth order terms in the

expansions of X, Y�� and Z (that is, they are constructedfrom ~g��, 0 and N0) while X, Y�� and Z are the termswhich contain linear contributions from h��, ’ and 7.Their specific forms are given in Appendix A. We cannow replace the new set of variables in the Israel matchingconditions and the scalar field boundary condition. At thefirst brane, �1, these take the form:

h0�� ~g��~g�8h0�8 �3

2N0UB� 0�7~g��

3

2N0@UB

@ 0’~g��

�3

2N0v1~g�� 25N0T

1��; (72)

and

’0 � N0@UB

@ 07� N0

@2UB

@ 20

’� N0@v1@ 0

; (73)

where, quantities like 0 and N0, not explicitly evaluatedat the boundary, must be evaluated at z � z1. Meanwhile,at the second brane, �2, the matching conditions are

h0�� ~g��~g�8h0�8�

3

2N0UB� 0�7~g��

3

2N0@UB

@ 0’~g��

�3

2N0v2~g�� 25N0T2

�� !!�20 ; (74)

and

’0 � N0@UB

@ 07� N0

@2UB

@ 20

’� N0@v2@ 0

; (75)

where again, quantities like 0 and N0 must be evaluatedat z � z2. Note that in the boundary conditions the new setof variables are linearly proportional to the energy-momentum tensor at the branes. In other words, devia-tions of the boundary conditions from the BPS stategenerate the existence of fields h��, ’ and 7 in thebulk, as expected in the low energy regime. In order tosolve the equations of motion it is useful to rewrite them

-10


in the form of a set of first order differential equations interms of the @z derivative. This can be done by definingnew fields :�� and ; to be sources of the fields h��, ’ and7, in the following equations:

h0�� ~g��~g�8h0�8 �

3

2N0UB� 0�7~g��

3

2N0@UB

@ 0’~g��

� :��; (76)

’0 � N0@UB

@ 07� N0

@2UB

@ 20

’ � ;: (77)

With these definitions of :�� and ;, it is found thatEqs. (65)–(67) can be reexpressed in a much simplerway. Respectively, these equations are found to be

2N0@UB

@ 0;�

1

3N0UB� 0�: �

N20

!20

X; (78)

�!4

0

N0:��

�0� 2N0!2

0Y��; (79)

��0!4

0

N0;�0� �0N0!2

0

�Z�

1

2�0X

�; (80)

where : � ~g��:��. Note that the left-hand sides ofEqs. (79) and (80) consist of first order derivatives of:�� and ; in terms of z, while the right-hand sidesdepend on second order derivatives in terms of thespace-time coordinate x. The boundary values of :��and ;, at the positions of the branes, can be computedwith the help of the boundary conditions (72)–(75). Theseare

:��z1� �32N0v1~g�� 2

5N0T1��; (81)

;�z1� � N0@v1@ 0

; (82)

at position z1, and

:��z2� �32N0v2~g�� 25N0T

2�� !!

�2; (83)

;�z2� � N0@v2@ 0

; (84)

at position z2. [Recall that !0�z1� � 1 and !0�z2� � !!.]We should not forget at this point the additional equationsfor the matter content at both branes. They come fromEq. (40) and, with the present notation, are given by

~r �T1�� 0; (85)

~r �T2��

1

!!��~r� !!�~g�8T2

�8 � 2�~r� !!�T2��; (86)

where covariant derivatives ~r� are constructed from the

064021

~g�� metric. Additionally, it is worth mentioning that theprojected Weyl tensor E�� and the loss parameter � canbe written in terms of the linear variables as

E�� !2

0

2N0

�:0��

1

3~g��~g

�8:0�8

��

1

4

!20

N0UB� 0�

�:��

1

3~g��~g�8:�8

��

3

8~g��

@UB

@ 0

;N0

�1

4

�1

4g��u� @� @� �

1

4g��@ �2

�4

Nr�r�N

�; (87)

� �1

N0

�;N0

�0: (88)

One of the main features of Eqs. (79) and (80) is thatnow they can be put in an integral form. That is, we canwrite

!40

N0

:��z��3

2~g��v1� 2

5T1��2

Z z

z1dzN0!2

0Y��; (89)

�0!4

0

N0;�z� � �� 1

0�@v1@ 1

0

�Z z

z1dz�0N0!

20

�Z�

1

2�0X

�;

(90)

where we have used the boundary conditions at the posi-tion of the first brane. To further proceed we must imple-ment a systematic expansion order by order in which thevacuum solution serves as the zeroth order solution. Wedevelop this in the next subsection.

B. Low energy regime expansion

Equations (89) and (90) are two integral equations ofthe system, which suggest the possibility of introducing asystematic expansion about the vacuum solution, order byorder. In the last subsection we saw that the matter contentof the branes are sources for the linear deviations h��, ’and 7 on the bulk. Therefore, it is sensible to study howthe bulk is affected by the matter distribution on thebrane, at different scales. In this way, let us considerthe following expansion for the energy-momentum tensorof the matter content on the branes, as well as for thesupersymmetry breaking potentials:

T1�� T1�0�

�� T1�1�� T1�2�

�� ; (91)

v1 � v�0�1 � v�1�1 � v�2�1 � � � � ; (92)

and

T2�� T2�0�

�� T2�1�� T2�2�

�� ; (93)

v2 � v�0�2 � v�1�2 � v�2�2 � � � � : (94)

-11


The parameter of the expansion is dictated by the scale atwhich each term becomes relevant in the physical problemof interest. Naturally, the expansion above will induce anexpansion of the source functions :�� and ;, given by

:�� :�1�� :�2�� ; (95)

; � ;�1� � ;�2� � � � � ; (96)

in this way, the boundary conditions (81)–(84), can nowbe written, order by order, as

:�i�1�� z1� �

32N0v

�i�1 ~g�� 25N0T

1�i�� ; (97)

;�i�1��z1� � N0@v�i�1@ 0

; (98)

at position z1, and

:�i�1�� z2� �

32N0v

�i�2 ~g�� 2

5N0T2�i�� !!�2; (99)

;�i�1��z2� � N0@v�i�2@ 0

; (100)

at position z2. Note the convention whereby the indices ofthe left-hand side are raised by one unit in relation tothose on the right-hand side. Continuing with the con-struction, we must also consider the expansion of h��, ’and 7:

h�� h�1�� h�2�� ; (101)

’ � ’�1� � ’�2� � � � � ; (102)

7 � 7�1� �7�2� � � � � : (103)

They are defined to satisfy the following first order dif-ferential equations with sources :�i�� and ;�i�:

h�i�0�� ~g��~g�8h�i�0�8 �

3

2N0UB� 0�7�i�~g��

�3

2N0@UB

@ 0’�i�~g��

� :�i��; (104)

’0�i� � N0

@UB

@ 07�i� � N0

@2UB

@ 20

’�i� � ;�i�: (105)

Finally, recall that the quantities X, Y�� and Z definedin Eqs. (71) depend on the fields h��, ’ and7. Therefore,we must consider the following expansion:

X � X0 � X�1� � X�2� � � � � ; (106)

Y �� Y0�� Y�1�

�� Y�2�� ; (107)

064021

Z � Z0 � Z�1� � Z�2� � � � � ; (108)

where the index ‘‘0’’denotes the dependence on the zerothorder quantities, N0, 0 and !0, and the index i � 1

denotes quantities that depend linearly on h�i��, ’�i� and

7�i�. The precise form of the expanded functions X�i�, Y�i��

and Z�i� are shown in Appendix A. Observe that it follows

from Eqs. (85) and (86) that the ith order term, Ta�i�� , inthe expansion of the energy-momentum tensor for thematter living in the brane, satisfies the following conser-vation relation:

~r �T1�i�� 0; (109)

~r �T2�i��

1

!!��~r� !!�~g�8T2�i�

�8 � 2�~r� !!�T2�i�� : (110)

With all these previous definitions we can now cast theequations of motion in the following way:

2@UB

@ 0;�i�1� �

1

3UB� 0�:�i�1� �

N0

!20

X�i�; (111)

�!4

0

N0:�i�1��

�0

� 2N0!20Y

�i��; (112)

��0!4

0

N0;�i�1�

�0

� �0N0!20

�Z�i� �

1

2�0X�i�

�: (113)

This last set of equations shows the desired low energyregime expansion. It states that :�i�1�

�� and ;�i�1� can be

solved in terms of the lower order quantities X�i�, Y�i��

and Z�i�. At the same time, the functions X�i�, Y�i�� and

Z�i� can be solved in terms of :�i�� and;�i�, as evident fromtheir definitions. Note that the former means that we cancompute E�� and � to any desired order, as indicated byEqs. (87) and (88). In this way, starting with the zerothorder solutions 0, N0 and ~g��, we can arrive at anydesired order for the full solutions to the bulk equations , N and g��. The present method is similar to the oneintroduced by Kanno and Soda for the Randall-Sundrummodel [37,38] as well as to other schemes [31,39–41] fordilatonic brane worlds.

We mentioned that the expansion parameter was dic-tated by the scale at which each term of the energy-momentum tensor expansion was relevant for the physicalprocess of interest. In terms of the bulk quantities, onefinds that the effect on the variation of the scale in theextra-dimensional direction is related to the variation ofthe scales in the space-time direction as follows:

h00�i�1� ’

N20

!20

�h�i�: (114)

-12


That is, second derivatives of the �i� 1�th order variablesin terms of z are of the same order as the second orderderivatives of the ith order variables in terms of space-time coordinates.

As in the previous case, we can rewrite Eqs. (112) and(113) in an integral form:

!40

N0

:�i�1��

3

2~g��v1 � 25T

1�� 2

Z z

z1dzN0!2

0Y�i��;

(115)

�0!4

0

N0;�i�1� � �1

@v1@ 1

0

�Z z

z1dz�0N0!2

0

�Z�i� �

1

2�0X�i�

�;

(116)

where �1 � �� 1�. This form of the expanded equationsof motion are highly relevant; they are the basis for thederivation of the effective theory developed in the nextsection.

Summarizing, in this section we have developed anexpansion procedure to solve the entire theory in a sys-tematic form. As we said, to solve this system of equa-tions we need to know the form of the zeroth order modulifields 1

0 and 20, and the metric ~g�� and, therefore, an

effective theory for these fields is required. In the follow-ing section, we are going to deduce the four-dimensionalequations of motion governing these functions, as well asthose for higher order terms in the expansion.

V. FOUR-DIMENSIONAL EFFECTIVE THEORY

In the previous section we have shown how to constructthe low energy regime as a consistent expansion. In thissection we see how to define the four-dimensional effec-tive theory at the position of the branes starting from theexpansion above. Concordant with the expansion, theeffective theory must be defined order by order.

A. General case

The effective equations governing the variables at theith order can be obtained by evaluating the integral equa-tions (115) and (116) at z � z2. In other words, the ithorder equations are given by

2Z z2

z1dzN0!

20Y

�i��

3

2� !!4v�i�2 � v�i�1 ~g�� 25�T

2�i�� !!2

�T1�i�� ; (117)

Z z2

z1dz�0N0!

20

�Z�i� �

1

2�0X�i�

�� 2 !!

4 @v�i�2

@ 20

� �1@v�i�1@ 1

0

;

(118)

where �2 � �� 2�. Additionally, another equation can beobtained by evaluating Eq. (111) at the boundary position

064021

z1. The resulting equation is

X�i��z1� � 2@UB

@ 10

@v�i�1@ 1

0

� 2UB� 10�v

�i�1

� 25

3UB�

10�~g

��T1�i�� : (119)

Note that a similar equation would have been obtained byevaluating (111) at the boundary position z2, however, thiswould not give a new equation of motion.Equations (117)–(119) are the desired equations of thefour-dimensional effective theory. Solving these equa-tions for the ith order variables allows us to obtain theeffective equations for the �i� 1�th order variables, aftercorrectly integrating (104) and (105). The form in whichEqs. (117)–(119) are presented is, at this level, abstractand it is difficult to appreciate the effective theory inmore familiar terms. In the next subsection we analyzein detail the effective theory at the zeroth order in theexpansion (i � 0).

B. Zeroth order effective theory

It will be particularly interesting to analyze the form ofthe effective equations at the zeroth order in more detail.In this case the metric at the first and second branes areconformally related. In terms of the treatment developedin the previous section, the effective theory is

Z z2

z1dzN0!

20Y

0��

3

4~g�� !!

4v�0�2 � v�0�1

� 252�T2�0�� !!2 � T1�0�

�� ; (120)

Z z2

z1dz�0N0!2

0

�Z0 �

1

2�0X0

�� 2 !!4 @v

�0�2

@ 20

� �1@v�0�1

@ 10

;

(121)

X0�z1� � 2@UB

@ 10

@v�0�1

@ 10

� 2UB� 10�v

�0�1

� 253UB�

10�~g

��T1�0�� : (122)

In the following, we shall omit the 0 index denoting thezeroth order terms. To rewrite this theory in more famil-iar terms, we need to integrate the terms on the left-handside of Eqs. (120) and (121). In order to do this we need toevaluate the terms at the boundaries. Hence it is particu-larly useful to express the theory in terms of the modulifields 1 and 2. It is then possible to obtain the followingeffective theory (Appendix B shows how to derive thenext results). The Einstein’s equations, obtained from(120), are found to be

-13


"2 ~G�� ~g�� ~�"2 � ~r�~r�"

2

�3

4�ab

�1

2~g��~g�8@� a@8 b � @� a@� b

�

�k 252

� !!2T2�� T1

�� 3

4~g��V; (123)

while the moduli fields equations, obtained fromEqs. (120) and (122), are found to be

~� a��~g��abc@� b@�

c�1

2�ac

@V@ c

�2

3�ac

@"2

@ c~R

�2

3k 25 !!�

ac @ !!@ c

~g��T2��: (124)

In the previous expressions, the index a labels the posi-tions 1 and 2. Additionally, note the presence of a confor-mal factor "2 in front of the Einstein’s tensor G��, givenby

"2 � kZ 2

1d

�@UB

@

��1!2; (125)

where ! is given by Eq. (47). The coefficient k is anarbitrary positive constant with dimensions of inverselength, which has been incorporated to make "2 dimen-sionless. The symmetric matrix �ab is a function of themoduli fields, that can be regarded as the metric of thespace spanned by the moduli in a sigma-model approach,with �ab its inverse. The elements of �ab are given by

�11 � ��21

�k

UB� 1��

1

2"2

�; (126)

�22 � ��22

!!2k

UB� 2�; (127)

�12 � ��11 ��1

2

!!2k

UB� 2�; (128)

with �21 � �12. Associated with the above metric, wehave defined a set of connections �abc. These are given by

�abc �1

2�ad

�@�bd@ c

�@�dc@ b

�@�bc@ d

�: (129)

Finally, we have also defined an effective potential Vwhich depends linearly on the supersymmetry breakingpotentials u, v1 and v2. This is defined as

V �k2

Z 2

1d

�@UB

@

��1!4u� 2k� !!4v2 � v1: (130)

064021

Finally, we cannot forget the matter conservation rela-tions. These take the form:

~r �T1�� 0; (131)

~r �T2��

1

!!��~r� !!�~g�8T2

�8 � 2�~r� !!�T2��: (132)

The fact that the energy-momentum tensor of the secondbrane does not respect the standard matter conservationrelation is due to the frame chosen to describe the system.

The generic form of the theory displayed by Eqs. (123)and (124) is of a biscalar tensor theory of gravity, with thetwo scalar degrees given by 1 and 2. The above set ofequations can be obtained from the following action:

S�1

k 25

Zd4x

��~g

p �"2 ~R�

3

4~g��ab@�

a@� b�

3

4V�

�S1��1; ~g��S2��2; !!2~g��: (133)

Equation (133) is an important result. It can be shown thatthe effective action (133) corresponds to the moduli-spaceapproximation [42], where the relevant fields of the theoryare just simply the free degrees of freedom of the vacuumtheory, promoted to be space-time dependent fields. Arelevant aspect of this theory, is that the equation ofmotion for a moduli field a will depend linearly on thetrace of the energy-momentum tensor of the brane a butnot on the one belonging to the opposite brane. Moreprecisely, it can be shown that (124) has the form:

~� 1 � �k 256

@UB

@ 1~g��T1

�� ; (134)

~� 2 � �k 256

@UB

@ 2 !!�2~g��T2�� : (135)

This means that, in the low energy regime, the modulifields are driven by the matter content of the branes(recall that the moduli are parametrizing the positionsof the branes). This behavior of the moduli fields isindependent of the frame choice (see next subsection).

C. Einstein’s frame

Note that in Eq. (133) the Newton’s constant dependson the moduli fields. This theory can be rewritten in theEinstein frame where the Newton’s constant is indepen-dent of the moduli. Considering the conformal transfor-mation:

~g �� ! g�� "2~g��; (136)

-14


we are then left with the following action:

S �1

k 25

Zd4x

��g

p�R�

3

4g��ab@�

a@� b �

3

4V�

�S1��1; A21g�� S2��2; A

22g��; (137)

where now the sigma-model metric �ab is given by

�11 � 2��21

k2A41

U2B�

1�

�1�

1

2kUB�

1�A�21

�; (138)

�22 � 2��22

k2A42

U2B�

2�

�1�

1

2kUB�

2�A�22

�; (139)

�12 � �2��11 ��1

2

k2A21A

22

UB� 1�UB�

2�: (140)

As usual, �ab is the inverse of �ab, and �cab are theconnections. It is possible to show that �ab is a positivedefinite metric. Additionally, we have defined the quan-tities A1 and A2 (which are functions of the moduli) to be

A�21 �"2�k

Z 2

1d

�@UB

@

��1

exp��1

2

Z

1��1d

�;

(141)

A�22 �"2 !!�2

�kZ 2

1d

�@UB

@

��1

exp��1

2

Z

2��1d

�: (142)

Also, the potential V is now found to be

V �k2"�4

Z 2

1d

�@UB

@

��1!4u� 2k�A4

2v2 � A41v1:

(143)

The present form of the theory can be further worked outto put the sigma model in a diagonal form with theredefinition of the moduli fields. However, in the presentwork we shall continue with the theory in the currentnotation. Let us finish this section by expressing theequations of motion in the Einstein frame. TheEinstein’s equations are

G�� 3

4�ab

�1

2g��g�8@� a@8 b � @� a@� b

�

�k 252

�A22T

2�� A2

1T1��

3

4g��V; (144)

and the moduli fields’ equations are

� 1 � �g��1bc�@�

b��@� c� �1

2�1c @V

@ c

�k 256

@UB

@ 1 A21g

��T1��; (145)

064021

� 2 � �g��2bc�@�

b��@� c� �1

2�2c @V

@ c

�k 256

@UB

@ 2 A22g

��T2��: (146)

Additionally, the matter conservation relations now read

r�T1��

1

A1��r�A1�g�8T1

�8 � 2�r�A1�T1��; (147)

r�T2��

1

A2��r�A2�g

�8T2�8� 2�r�A2�T

2��: (148)

In the next sections, we are going to study this theoryin more detail, analyzing a few examples (including theRandall-Sundrum case) and studying the cosmologicalevolution of the moduli for late cosmological eras. Weshall pay special attention to the observational constraintsthat can be imposed on this model.

D. A few examples

In this subsection we shall briefly analyze two well-known cases, namely, the exponential case UB� � �V0e� (dilatonic brane worlds), and the Randall-Sundrum case, which can be derived as particular casesof the formalism described above.

1. Exponential case

This case is particularly simple, and has been studiedin detail in several previous works [25–31]. The potentialto consider is UB� � � V0e� , with � being a constantparameter of the theory. Here, it is possible (and conve-nient) to parametrize the theory using a new set of fields.Let us assume that V0 > 0 and consider the followingtransformations:

e2#1cosh2#2 �2

1� 2�2 e��1�2�2� 1=2�;

e2#1sinh2#2 �2

1� 2�2 e��1�2�2� 2=2�:

(149)

If V0 < 0 then we should interchange 1 and 2 in theprevious definition of #1 and #2. Inserting this notationback in the theory, it is possible to obtain the followingeffective action:

S �1

k 25

Zd4x

��g

p�R�

12�2

1� 2�2 �@#1�2 �

6

1� 2�2

�@#2�2 �

3

4V�� S1��1; A

21g��

�S2��2; A22g��; (150)

where now the coefficients A1 and A2 are functions of thefields #1 and #2, given by

-15


A21 � e�2#1

V0

k

�1

2�1� 2�2�e2#1cosh2#2

�1=�1�2�2�

;

A22 � e�2#1

V0

k

�1

2�1� 2�2�e2#1sinh2#2

�1=�1�2�2�

:

(151)

This form of the theory was already obtained in [30], inthe moduli-space approximation approach, and extensiveanalysis to it have been made.

2. Randall-Sundrum case

Finally, it is also instructive to examine the equationsfor the Randall-Sundrum case. This can be obtained as alimiting case of the previous example, by letting �! 0.The only field surviving the limiting process is #2, andthe effective action is

S �1

k 25

Zd4x

��g

p�R� 6�@#2�

2 �3

4V�

� S1��1; A21g�� S2��2; A

22g��; (152)

where now the coefficients A1 and A2 are functions of thefield #2 alone, and are given by

A21 �

V0

2kcosh2#2; A2

2 �V0

2ksinh2#2: (153)

VI. LATE-TIME COSMOLOGY

A simple and important application of the formalismdeveloped above is the study of cosmological solutions forthe most recent epoch of the Universe. In simple terms,current observations reveal that the moduli fields are notrelevant for the present evolution of the Universe and theirdynamics is strongly suppressed. In this section we shallanalyze the possibility of stabilizing the moduli fields 1

and 2. Interestingly, it will be found that, in order tostabilize the system, constraints must be placed on theglobal configurations of branes, particularly, on the classof possible potentials and also on the position of thebranes with respect to the background.

A. Cosmological equations

Let us derive the cosmological equations of motion ofthe present system, valid for a flat, homogeneous andisotropic universe. We can derive the equations using theflat Friedmann-Robertson-Walker metric:

ds2 � g��dx�dx� � �dt2 � a2�t�'ijdx

idxj: (154)

Here, g�� corresponds to the Einstein’s frame metric,conformally related to the physical metric, ~g��, of thebrane frame. Using the metric (154), jointly with theeffective action (137), we are left with the followingFriedmann equations:

064021

'aa� �

k 2512

A41��1 � 3p1� �

k 2512

A42��2 � 3p2�

�1

4�ab _ a _ b �

1

4V; (155)

H2 �k 256A41�1 �

k 256A42�2 �

1

8�ab _ a _ b �

1

4V; (156)

where V is given in Eq. (143) and H � _a=a is the Hubbleparameter. The moduli fields equations, on the other hand,are given by

' 1 � 3H _ 1 � ��1ab

_ a _ b �1

2�1c @V

@ c

�1

6k 25

@UB

@ 1 A41��1 � 3p1�; (157)

' 2 � 3H _ 2 � ��2ab

_ a _ b �1

2�2c @V

@ c

�1

6k 25

@UB

@ 2 A42��2 � 3p2�: (158)

Additionally, the matter conservation relations can beread as

_� 1 � 3H��1 � p1� � �3_A1

A1��1 � p1�; (159)

_� 2 � 3H��2 � p2� � �3_A2

A2��2 � p2�: (160)

The above set of equations are the corresponding equa-tions of motion describing the cosmological behavior ofthe brane system. Since the sigma-model metric is posi-tive definite, the moduli fields are not going to result in anaccelerating universe. Thus, in the present context, theonly way to obtain an accelerating universe is to considersupersymmetry breaking potentials. A more detailedanalysis of this system, taking into account the fulldependence on the extra kinetic terms coming from thesigma-model formulation is complicated, and shall beomitted in the present work. However, since we are inter-ested mostly in the low energy regime of branes, we shallstudy the case in which the moduli are slowly evolvingcompared to the Hubble parameter. We examine this inthe next subsection.

B. Late-time cosmology

Let us consider the case in which the moduli fields areslowly evolving compared with the Hubble parameter,

-16


that is j _ 1j, j _ 2j � H, and that the supersymmetrybreaking potentials are such that u; v1; v2 � 0. Then,the Friedmann equations of the system are

'aa� �

k 2512

A41��1 � 3p1� �

k 2512

A42��2 � 3p2�; (161)

H2 �k 256A41�1 �

k 256A42�2: (162)

The moduli fields equations, on the other hand, are givenby

' 1 � 3H _ 1 � �1

6k 25

@UB

@ 1 A41��1 � 3p1�; (163)

' 2 � 3H _ 2 � �1

6k 25

@UB

@ 2 A42��2 � 3p2�: (164)

And the matter conservation relations can now be writtenas

_� 1 � 3H��1 � p1� � 0; (165)

_� 2 � 3H��2 � p2� � 0: (166)

It is possible to appreciate that the moduli fields aredriven by the matter content of the branes, while theevolution of the Universe remains unaffected by themoduli. In particular, if the Universe is matter dominated,then there is an attractor for the moduli towards theextremes of the supersymmetric potential UB. To bemore precise, 1 will be attracted to the minimum ofUB, while 2 will be attracted to the maximum of UB.This is sketched in Fig. 4. It is therefore sensible toassume that the present state of the Universe is very closeto this configuration, where the potential is being extre-mized by the moduli. In the next subsection we assume

FIG. 4. The figure shows the case when the branes are matterdominated. The moduli fields 1 and 2 are driven to theminimum and maximum of the superpotential UB, respec-tively.

064021

that this is the case in order to place observational con-straints on the model.

C. Observational constraints

We now turn to the task of placing observationalbounds on the theory. In the last subsection we observedthat the branes are cosmologically driven by the mattercontent in them. In particular, the evolution of the moduliis such that they extremize the value of the supersym-metric potential UB. Therefore we expect that at latecosmological eras, such as the present era, the branesare far apart and their tensions nearly at their extremevalues. This corresponds to an interesting global condi-tion on the configuration of the system, however, it can beconstrained by current observations. For example, we cancompute the post-Newtonian-Eddington coefficient �which is constrained by measurements of the deflectionof radio waves by the sun to be � � 1� �2:1� 2:3� 10�5 [43–45]. The parameter � is defined as

1� � � 2B2

1� B2 ’ 2B2; (167)

where B2 � �abBaBb, and Ba is defined as

Ba �1

A1

@A1

@ a: (168)

Observe that Ba is constructed from A1, which is thefactor that appears in the action for the matter contentat the �1 brane. This is because we are assuming, withoutloss of generality, that we are performing measurementsfrom the �1 brane. The quantity B2 can be computedexactly:

B2 �1

8

�1�

1

2kUB�

1�A�21

�: (169)

Then, we finally obtain

1� � ’2

3

�1�

1

2kUB�

1�A�21

�: (170)

(Note that this result is independent of k.) Recall that A1 isrelated to an integral over the entire bulk. This is a veryimportant result: observational measurements constrainthe global configuration of the brane system. For example,we can compute the � parameter for the exponentialpotential. In this case, we obtain

1��’4

3

�2

1�2�2�2

3

1

1�2�2e�1�2�2�� 1� 2�=2�; (171)

which agrees with earlier computations [30]. If the branesare such that the moduli are extremizing the potentialUB,then we have �� 2 � 1� ! �1, and the second termwill disappear in (171). Then, we end up with a constraint

-17


on �, the parameter of the model:

1� � ’4

3

�2

1� 2�2 ; (172)

which gives j�j< 1:25 10�3. Interestingly, this bounddoes not affect the global configuration of the brane, butonly the parameter of the theory. This is certainly aproblem since theoretically viable values for� are usuallyof the order of unity. In the Randall-Sundrum case there isno potential, and the moduli cannot be driven to anystable configuration. In this case, the constraint takesthe form:

1� � ’2

3tanh2#2: (173)

Thus we see that in the Randall-Sundrum case a stabili-zation mechanism for the radion field is necessary inorder to agree with observations.

VII. CONCLUSIONS

In this paper we have studied several aspects of the lowenergy regime of BPS-brane-world models. We have de-veloped a systematic procedure to obtain solutions to thefull system of equations, consisting in a linear expansionof the fields about the static vacuum solution of thesystem. As a result, it was shown that five-dimensionalsolutions can be obtained at any desired order in theexpansion. In particular, the projected Weyl tensor E��and the loss parameter � can be computed with anydesired accuracy. Additionally, and probably more impor-tant, an effective four-dimensional system of equationshas been obtained. Concordant with the method, thesefour-dimensional effective equations must be consideredup to the desired order in the expansion. For instance, wehave analyzed in detail the zeroth order effective theory,which agrees with the moduli-space approximation [42].At this order, the metrics of both branes are conformallyrelated, and the complete theory corresponds to a biscalartensor theory of gravity [Eq. (137)]. The approach fol-lowed in this paper to obtain the effective theory issimilar to the one followed in a previous work [31], wherethe cosmological setup of the case UB / e� was inves-tigated. However, the treatment in this article was moregeneral in two senses: we did not restrict the form of theBPS potential, and we deduced the complete Einstein’sequations governing the system.

We have also seen that the moduli fields—as defined inthe zeroth order part of the theory—can be stabilizedwith the help of effective couplings between the moduliand matter that arise naturally. This result comes from thefact that the equations of motion for the projected bulk

064021

scalar field on the brane depend only on the energymomentum of the respective brane. This allows satisfac-tory late cosmological configurations and enables us toplace observational constraints on the model. An impor-tant result was the computation of the Post NewtonianEddington coefficient � in terms of the moduli[Eq. (170)]. This result is relevant since � is currentlyone of the most constrained parameters of general rela-tivity. For example, in the exponential case it was foundthat j�j< 1:25 10�3.

Our results are applicable to many other aspects ofbrane worlds not considered in this paper. For example,the development of the theory at the first order in theperturbed metric would allow an ideal background tostudy the cosmological perturbations and the cosmicmicrowave background predictions [46,47]. Add-itionally, many results where the exponential case UB /e� was considered can now be extended to the generalcase of an arbitrary potential.

ACKNOWLEDGMENTS

We are grateful to Philippe Brax and Carsten van deBruck for useful discussions. This work is supported inpart by PPARC and MIDEPLAN (G. A. P.).

APPENDIX A: DEFINITION OF X, Y�� AND Z

Here we define the zeroth order and the linear expan-sions of X, Y�� and Z of Eqs. (71). The zeroth orderquantities are

X0 � �@ 0�2 � 4

3~R� 8!�1

0~�!0 �!2

0u; (A1)

Y0�� ~G��

3

4

�1

2~g��@ 0�

2 � @� 0@� 0

�� N0!

20�

�1

�~g�� ~��N0!20� �

~r�~r��N0!2

0�

�3~g��@�!0@��N0!0� � 3@�!0@��N0!0�

�3@�!0@��N0!0� �3

8!2

0~g��u; (A2)

Z0 � �1

N0~g�� ~r��N0@� 0� � 2!�1

0 ~g�C@C!0@� 0

�!2

0

2

@u@ 0

: (A3)

Meanwhile, the linear terms in the expansions of X, Y��

and Z are given by the following expressions:

X � 2@ 0@’� 43�~r�

~r�h�� ~�h� � 4!�1

0 ~g�8~g�C

�~r�hC8 � ~r8h�C � ~rCh�8�~r�!0; (A4)

-18


Y�� 1

2�~r�

~r�h�� ~r�~r�h�� ~r�

~r�h� ~�h�� ~g�� ~r�~r8h�8 � ~g�� ~�h� �

1

2�N0!2

0��1~g�C

�~r�hC� � ~r�h�C � ~rCh�� 2~g�� ~r�h�C � ~g�� ~rCh�~r��N0!20�

�3

4�~g��~g

�8@� 0@8’� @� 0@�’� @� 0@�’� � ~g�� ~�7� ~r�~r�7� ~g��!

�10 ~g�8@�!0@87

�!�10 �@�!0@�7� @�!0@�7 �

1

N0�2~g��~g

�8@�N0@87� @�N0@�7� @�N0@�7�; (A5)

Z� 12~g�C~g��~r�hC�� ~r�h�C� ~rCh��@� 0

�~g�� ~r�7~r� 0� ~�’�2!�10 ~g�C@C!0@�’: (A6)

To compute the ith order terms of the linearexpansions, with i > 0, it is enough to add an index‘‘i’’ to every linear variable in the expressionsabove.

APPENDIX B: COMPUTATION OF THE ZEROTHORDER THEORY

Here we indicate how to compute the integralin the left-hand side of Eq. (120). The integralpresent in Eq. (121) can be solved in a similarway. First, it is important to note the following iden-tity:

@��N0!0� � �N0�0!0@� 0 �4

UB�@�!0�

0: (B1)

Additionally, recall that we can parametrize the z coor-dinate using the monotonic zeroth order solution for thescalar field 0�z�. That is, we can write

064021

dz �1

N0

�@UB

@ 0

��1d 0: (B2)

Using the previous equations, with boundaries 0�z1�� 1 and 0�z2�� 2, then the mentioned integral can besolved as

kZ z2

z1dzN0!2

0Y0��"2 ~G�� ~g�� ~�"2� ~r�

~r�"2� ~g��

3k8

Z 2

1d

�@UB

@

��1!4u�

3

4�ab

�1

2~g��~g

�8@� a@8

b�@� a@�

b�;

(B3)

where "2 and �ab are defined as in Sec. V. To finish, weshould mention that in obtaining this result it was usefulto notice that "2 does not only depend on 1 and 2

through the integration limits in (125), but also through!2 present in the integrand. Recall that ! is normalizedto be 1 at the position of the first brane, and thereforedepends on 1. This in turn means that the space-timederivative of "2 will have the form:

@�"2��1

1

�1

2"2�

k

UB� 1�

�@�

1��12

!!2k

UB� 2�@�

2:

(B4)

[1] P. Horava and E. Witten, Nucl. Phys. B460, 506(1996).

[2] P. Horava and E. Witten, Nucl. Phys. B475, 94 (1996).[3] A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram,

Nucl. Phys. B552, 246 (1999).[4] K. Akama, Lect. Notes Phys. 176, 267 (1982).[5] V. A. Rubakov, Phys. Lett. 125B, 136 (1983).[6] M. Visser, Phys. Lett. 159B, 22 (1985).[7] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G.

Dvali, Phys. Lett. B 436, 257 (1998).[8] I. Antoniadis, Phys. Lett. B 246, 377 (1990).

[9] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999).

[10] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690(1999).

[11] N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, and R.Sundrum, Phys. Lett. B 480, 193 (2000).

[12] S. Forste, Z. Lalak, S. Lavignac, and H. P. Nilles, Phys.Lett. B 481, 360 (2000).

[13] Ph. Brax and C. van de Bruck, Classical QuantumGravity 20, R201 (2003).

[14] D. Langlois, Prog. Theor. Phys. Suppl. 148, 181 (2003).

-19


[15] R. Maartens, Living Rev. Rel. 7, 1 (2004).[16] Ph. Brax, C. van de Bruck, and A. C. Davis, hep-th/

0404011.[17] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math.

Phys.) K1, 966 (1921); O. Klein, Z. Phys. 37 895(1926).

[18] C. D. Hoyle et al., Phys. Rev. Lett. 86, 1418 (2001).[19] E. E. Flanagan, S. H. H. Tye, and I. Wasserman, Phys. Rev.

D 62, 044039 (2000).[20] C. van de Bruck, M. Dorca, C. J. A. P. Martins, and M.

Parry, Phys. Lett. B 495, 183 (2000).[21] W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. 83,

4922 (1999).[22] J. Lesgourgues and L. Sorbo, Phys. Rev. D 69, 084010

(2004).[23] E. Bergshoeff, R. Kallosh, and A.V. Proeyen, J. High

Energy Phys. 10 (2000) 033.[24] P. Binetruy, J. M. Cline, and C. Grojean, Phys. Lett. B

489, 403 (2000).[25] Ph. Brax and A. C. Davis, Phys. Lett. B 497, 289 (2001); J.

High Energy Phys. 05 (2001) 007.[26] C. Csaki, J. Erlich, C. Grojean, and T. J. Hollowood,

Nucl. Phys. B584, 359 (2000).[27] D. Youm, Nucl. Phys. B596, 289 (2001).[28] E. E. Flanagan, S. H. H. Tye, and I. Wasserman, Phys.

Lett. B 522, 155 (2001).[29] O. DeWolfe, D. Z. Freedman, S. S. Gubser, and A. Karch,

Phys. Rev. D 62, 046008 (2000).[30] Ph. Brax, C. van de Bruck, A. C. Davis, and C. S. Rhodes,

Phys. Rev. D 67, 023512 (2003).

064021

[31] S. Kobayashi and K. Koyama, J. High Energy Phys. 12(2002) 056.

[32] A. Lukas, B. A. Ovrut, K. Stelle, and D. Waldram, Phys.Rev. D 59, 086001 (1999).

[33] T. Damour amd G. Esposito-Farese, Classical QuantumGravity 9, 2093 (1992).

[34] T. Damour, Phys. Rev. Lett. 70, 2217 (1993).[35] Ph. Brax and N. Chatillon, J. High Energy Phys. 12

(2003) 026.[36] G. A. Palma, Ph. Brax, A. C. Davis, and C. van de Bruck,

Phys. Rev. D 68, 123519 (2003).[37] S. Kanno and J. Soda, Phys. Rev. D 66, 083506 (2002).[38] S. Kanno and J. Soda, Astrophys. Space Sci. 283, 481

(2003).[39] S. Mukohyama and L. Kofman, Phys. Rev. D 65, 124025

(2002).[40] S. Mukohyama and A. Coley, Phys. Rev. D 69, 064029

(2004).[41] S. Kanno and J. Soda, Gen. Relativ. Gravit. 36, 689

(2004).[42] Gonzalo A. Palma and Anne-Christine Davis, hep-th/

0407036.[43] T. M. Eubanks et al., Am. Phys. Soc., Abstract K 11.05

(1997).[44] C. Will, Living Rev. Rel. 4, 4 (2001).[45] B. Bertotti, L. Iess, and P. Tortora, Nature (London) 425,

374 (2003).[46] C. S. Rhodes, C. van de Bruck, Ph. Brax, and A. C. Davis,

Phys. Rev. D 68, 083511 (2003).[47] K. Koyama, Phys. Rev. Lett. 91, 221301 (2003).

-20

Low energy branes, effective theory, and cosmology

Documents

Transcript of Low energy branes, effective theory, and cosmology