26 July 2001

Physics Letters B 513 (2001) 156–162www.elsevier.com/locate/npe

Brane cosmology and naked singularities

Ph. Braxa,1, A.C. Davisb

a Theoretical Physics Division, CERN CH-1211 Geneva 23, Switzerlandb DAMTP, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WA, UK

Received 25 May 2001; accepted 9 June 2001Editor: P.V. Landshoff

Abstract

Brane-world singularities are analysed, emphasizing the case of supergravity in singular spaces where the singularity puzzleis naturally resolved. These naked singularities are either time-like or null, corresponding to the finite or infinite amount ofconformal time that massless particles take in order to reach them. Quantum mechanically we show that the brane-world nakedsingularities are inconsistent. Indeed we find that time-like singularities are not wave-regular, so the time-evolution of wavepackets is not uniquely defined in their vicinity, while null singularities absorb incoming radiation. Finally, we stress that forsupergravity in singular spaces there is a topological obstruction, whereby naked singularities are necessarily screened off bythe second boundary brane. 2001 Elsevier Science B.V. All rights reserved.

1. Introduction

Brane cosmology [1] has recently appeared as aframework where thorny issues such as the cosmologi-cal constant problem can be tackled. In particular, newmechanisms have been proposed whereby the vacuumenergy of the brane-world curves the fifth dimensionleaving a flat four-dimensional brane-world intact [2].Unfortunately, this scenario seems to fail as the exis-tence of a naked singularity in the bulk prevents onefrom obtaining a smooth five-dimensional space-time.This singularity needs to be resolved leading to a fine-tuning between the tension on the brane-world and ofthe ghost brane sitting at the singularity [3].

Another proposal involves supergravity in singularspaces [4]. In that caseN = 2 supergravity lives in the

E-mail addresses: philippe.brax@cern.ch (Ph. Brax),a.c.davis@damtp.cam.ac.uk (A.C. Davis).

1 On leave of absence from Service de Physique Théorique,CEA-Saclay F-91191 Gif/Yvette Cedex, France.

bulk and is broken on the brane-world. The breakingof supersymmetry ensures that a non-static configura-tion is generated [5]. The resulting brane-world met-ric is of the FRW type with an acceleration parameterq0 = −4/7 and an equation of stateω= −5/7, withinthe experimental ball-park [6]. Unfortunately, a thor-ough analysis of the coupling of supergravity in singu-lar spaces to ordinary matter on the brane-world showsthat the amount of supersymmetry breaking needs tobe fine-tuned to the level of the critical energy densityof the universe [7]. Nevertheless, this model is rele-vant as it realizes in an explicit way a quintessencescenario [8] in five dimensions. As in the self-tunedbrane scenario there is a would-be singularity in thebulk. One of the aims of this Letter is to provide a de-scription of how a topological obstruction prevents theexistence of such a naked singularity in the bulk.

In a first section we shall study the classical trajec-tories of massive and massless particles by studyingthe geodesics in warped geometries with naked sin-gularities. For massless particles the singularity can

0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)00734-1

Ph. Brax, A.C. Davis / Physics Letters B 513 (2001) 156–162 157

be reached in either a finite or an infinite amount ofconformal time corresponding to time-like or null sin-gularities. The former corresponds to the self-tunedbrane while the latter appears for supergravity in sin-gular spaces. We then study the quantum mechanics ofgravitons and show that the time-like naked singulari-ties arerepulsons [9], repelling all incoming radiation.We also find that they are not wave-regular [10–12],i.e., the time evolution of wave packets is not uniquelydefined, prompting the necessity of imposing an ap-propriate boundary condition at the singularity, i.e.,knowing enough about its possible resolutions. On thecontrary, null singularities are wave-regular but absorbincoming radiation. In both case this signals a quan-tum inconsistency of the models. Fortunately, in thesupergravity case, the would-be singularity is absentdue to a topological obstruction associated with thepresence of four-forms in the bulk. This obstructionis reminiscent of the tadpole cancellation mechanismin string theory [13,14].

2. Geodesic motion

In this section we are interested in the classical mo-tion of massless and massive particles in the vicinityof the naked singularities arising in brane-world sce-narios. More precisely we consider the bulk geometryto be described by the metric

(1)ds2 = a2(u)(du2 − dη2 + dxi dx

i)

in conformal coordinates. The behaviour of the scalefactor a(u) is given by a power law close to thesingularity

(2)a(u)=(u

u0

)β

.

The curvature vanishes at the origin forβ >−1 and atinfinity for β <−1. We will investigate both situationsin the following.

The classical motion is characterized by the point-particle Lagrangian [15]

(3)L ≡ ds2

dτ2 = a2(u)(u2 − η2 + xi xi

).

The time and space independence of the Lagrangianleads to the conservation of energy and momentum

(4)E = −2a2(u)η, pi = 2a2(u)xi

thus giving the reduced Lagrangian

(5)L= a2(u)

(u2 + p2 −E2

4a4

).

The trajectories are determined by the constraint

(6)L= ε,

whereε = 0 for light-like paths andε = −1 for time-like paths. Let us consider a particle sitting initiallyon the brane-world with speedu|0. We can rewrite theLagrangian constraint as

(7)u2 + ε − u|20a4 − ε

a2 = 0.

This is the classical motion of a massive particle withzero total energy in the potential

(8)V (u)= ε − u|20a4(u)

− ε

a2(u).

First of all notice that the particles with a vanishinginitial velocity u0 = 0 remain on the brane.

Let us now consider the geodesics obtained bylaunching the particles from the brane to the singular-ity. Forβ > 0 the potential goes to minus infinity at thesingularity and vanishes at infinity. For−1 < β < 0the potential vanishes at the singularity and goes to mi-nus infinity at infinity. Forβ < −1 the potential goesto minus infinity at the singularity. There is a criticalpoint at

(9)a2∗ = 2+ 4u∣∣20

in the massive case. Forβ > 0 the critical point isbeyond the brane-world located atu0 while for β < 0it is between the brane-world and the origin.

Let us consider massive particles first. In the caseβ > 0, as the total energy vanishes and the potentialenergy of the particle is always negative, we find thatmassive particles are attracted by the singularity. In thecase−1< β < 0 massive particles evolve in the bulkbefore reaching a turning point for

(10)a2tp = 1+ u2

0.

Notice that this ensures that massive particles neverreach the singularity in the−1 < β < 0 case. Forβ < −1 the massive particles are attracted by thesingularity.

158 Ph. Brax, A.C. Davis / Physics Letters B 513 (2001) 156–162

Massless particles are attracted by the singularity.Indeed, the geodesics are given by

(11)u= u0 − 2u0

Eη,

showing that the singularity is reached in a finiteamount of conformal time forβ >−1, correspondingto a time-like singularity whereas forβ < −1 theamount of conformal time is infinite, i.e., a nullsingularity.

Two particularly relevant cases have been discussedlately. First of all, the self-tuned brane scenario [2] issuch thatβ = 1/3. This corresponds to an attractivetime-like singularity. All kinds of matter, whethermassive or massless, are attracted and reach thesingularity in a finite amount of conformal time. Assuch this singularity does not make sense classically. Ithas been argued that it needs to be resolved by puttingan appropriate brane located at the singularity whosetension compensates for the tension of the originalbrane atu0.

Another scenario invokes the presence of supergrav-ity in the bulk with broken supersymmetry on thebrane [5]. There is a would-be singularity whose exis-tence will be further discussed in Section 4. It is char-acterized byβ = −3/2. This corresponds to a null sin-gularity.

In the next section we will study the quantummechanical behaviour of massless particles in thevicinity of such naked singularities.

3. Wave-regularity of naked singularities

We have seen that the brane-world singularities at-tract massless particles. It is then relevant to analysetheir quantum mechanical behaviour in the neighbour-hood of the naked singularity. We shall restrict our at-tention to massless particles in the s-wave channel. Weassume that the only massless particle propagating inthe bulk is the graviton. We are interested in gravitonspolarized along the brane-world. The graviton wavefunction can be written as

(12)hij =H(u,x)εij ,

whereH(u,x) = H(u)eik.x , εij is the polarizationtensor andk is in the time direction. In the Einsteinframe the graviton equation reduces to the Laplace

equation

(13)�hij = 0.

The polarization tensor must be tracelessηij εij = 0and transverse tok implying thatε0i = 0. Denoting byε the spatial part of the polarization tensor we find abasis of these tensors given by off-diagonal symmetricmatrices withεab = εba = 1 and zero otherwise alongwith diagonal matrices such thatεaa = 1, εbb = −1,a < b. The latter are diagonal polarizations while theformer are transverse polarizations. In the diagonalcase put

(14)H(u,x)= aφ(u, x).

Then the scalar fieldφ satisfies the free wave equation

(15)∇µ∇µφ = 0.

In the transverse case the functionH(u,x) satisfies thefree scalar equation too.

In the following we shall concentrate on the scalarwave equation in five dimensions. It is particularlyuseful to introduce

(16)φ = a−3/2ψ,

which satisfies the Schrodinger equation

(17)ψ ′′ − Vψ = 0,

where

(18)V = −ω2 + (a3/2)′′

a3/2 .

It is possible to recast the Schrodinger equation intothe form [12]

(19)(�QQ−ω2)ψ = 0,

where

(20)

Q= − d

du+ 1

2

d lna3

du, �Q= d

du+ 1

2

d ln a3

du.

The Hamiltonian �QQ is a symmetric operator(f,Q�Qg)= (Q�Qf,g), where

(21)

(f, g)=∫duf ∗(u)g(u)+

∫duDuf

∗(u)Dug(u),

for functions depending only onu if one restricts thedomain of�QQ to the infinitely differentiable functions

Ph. Brax, A.C. Davis / Physics Letters B 513 (2001) 156–162 159

with compact support. Following [11] we choose a So-bolev norm as it is related to the energy of the scalarfield φ. In particular, fields with finite norm have finiteenergy. With this choice the Hamiltonian is symmetricbut is not guaranteed to be a self-adjoint operator[10,11]. Notice too that the Hamiltonian is a positiveoperator with two zero modes

(22)ψ1(u)= a3/2, ψ2(u)= a3/2

u∫dv

a3(u).

The rest of the spectrum is positive preventing theexistence of tachyons.

We can solve the Schrodinger equation correspond-ing to the brane-world singularities as

(23)V = −ω2 + 3β

2

(3β

2− 1

)1

u2.

The self-tuned brane scenario withβ = 1/3 and thesupergravity scenario withβ = −3/2 lead to attractivesingularities as the potential decreases at the originin the former case, and at infinity in the latter case,respectively. It is convenient to definez = ωu. Thegeneralized eigenstates read

ψ1ω(z)= √

z J(3β−1)/2(z),

(24)ψ2ω(z)= √

z J(1−3β)/2(z),

for β = 1/3. In the latter case, i.e., for the self-tunedbrane scenario, the solutions are expressed in terms ofzeroth order Bessel and Neumann functions

(25)ψ1ω(z)= √

z J0(z), ψ2ω(z)= √

zN0(z).

Close to the time-like singularity the constant termin the potential is negligible implying that all thesolutions behave like the two zero modes

(26)ψ1(z)= z3β/2, ψ2 = z1−3β/2,

for β = 1/3, and

(27)ψ1(z)= √z, ψ2(z)= √

z ln z,

for β = 1/3. As none of the eigenstates are oscillatoryin the neighbourhood of the time-like singularity thisimplies that no flux reaches it. This is natural whenthe singularity is repulsive. For attractive singularitiesthis is due to the extreme steepness of the potential.Such singularities arerepulsons [9]. For β < −1the singularity is at infinity where the eigenfunctions

behave like plane waves. This implies that in a scat-tering experiment there will be some absorption by thenull singularity.

We can now study whether the time evolution ofwave packets is well defined in the vicinity of thenaked singularities. To do that let us write the masslessKlein–Gordon equation in the form

(28)∂2φ

∂t2= −Mφ,

whereM is a second order partial differential operatordepending only on the spatial derivatives. After achange of variable,M reduces to the Hamiltonian�QQ. The Klein–Gordon equation defines a uniquetime evolution provided it can be written in theSchrodinger form

(29)∂φ

∂t= iM1/2φ

for a unique self-adjoint operatorM1/2. This is equiv-alent to finding a unique self-adjoint extension to thesymmetric operator�QQ, i.e., the Hamiltonian�QQ isessentially self-adjoint.

For null singularities at infinity, there is a singleself-adjoint extension of the symmetric operator�QQacting on functions decreasing fast enough at infin-ity [10]. Hence the time-evolution of wave packets iswell-defined. For the time-like case there is a usefulcriterion of essential self-adjointness [10,11]. Let usconsider the eigenvalue problem

(30)�QQψ = ±iψ.It reduces to a Schrodinger problem in a complexpotential

(31)V = V ± i.

Denote byn± the number of normalizable solutionsto (30). As the operator�QQ is real one hasn+ = n−,implying that there always exists self-adjoint exten-sions. Now the operator is essentially self-adjoint pro-vided n± = 0, i.e., the solutions are not normaliz-able. Due to the finiteness of the fifth dimension, theonly possible source of divergence is at the singularity.Therefore one must check whether or not the solutionsof (30) are normalizable close to the singularity.

In our case notice that in the vicinity of the singu-larity the extra complex term toV is negligible, im-plying that the solutions are expressed in terms of the

160 Ph. Brax, A.C. Davis / Physics Letters B 513 (2001) 156–162

two zero modes.2 The issue of the quantum mechan-ical behaviour of the singularity is now dependent onthe norm of these eigenfunctions. Using the fact thatthe covariant derivative of the metric vanishes, we findthat the norm ofψ1 is finite provided

(32)∫dua3 <∞,

which leads to

(33)β >−1

3.

Similarly the norm ofψ2 is finite provided

(34)∫du

1

a3 <∞,

leading to

(35)β <1

3.

Therefore, we find that there is always one of thezero modes which is normalizable. This implies thatthe Hamiltonian is not essentially self-adjoint. Hencewe cannot define a unique evolution operator in theneighbourhood of the singularity. Uniqueness of theevolution operator can be achieved if a physical choiceof boundary condition at the singularity is imposed.This requires more knowledge about the physics of thesingularity, i.e., its resolution.

We have thus shown that the quantum mechanicalbehaviour of brane-world singularities is pathological.Indeed the time-like singularities are repulsons whilenot wave-regular. This requires knowledge about theresolution of the singularity in order to define the timeevolution of wave packets in their vicinity. On thecontrary, null singularities are wave regular allowingone to study the evolution of wave packets in theirvicinity irrespective of the physical nature of thesingularity. Unfortunately, the null singularities have anon-vanishing absorption cross section which needs tobe interpreted in order to make sense. In particular, thisabsorption might signal the presence of fields at thenull singularity to which the gravitons couple. In anycase this requires a deep understanding of the natureof the singularity. The time-like case is exemplified by

2 The caseβ = 2/3 is special as the potential vanishes. It is easyto see that the eigenfunctions are normalizable.

the self-tuned brane models while the null case occursfor supergravity in singular spaces. In the next sectionwe will study the resolution of naked singularities insingular space supergravity.

4. Supergravity in singular spaces

We have seen that the brane-world naked singularityare either not wave-regular, so that the time evolutionoperator is not well-defined in their neighbourhood, orthey absorb incoming radiation. This is an inconsis-tency of the models which needs to be cured. In thefollowing we shall treat the case of supergravity in sin-gular spaces [4] where such singularities may occur.Nevertheless, we will show that there is a topologicalobstruction to the presence of naked singularities inthe bulk.

Let us first discuss the on-shell bosonic part of theLagrangian

(36)

Sbulk = 1

2κ25

∫ √−g5

(R − 3

4

(gij ∂µφ

i∂µφj + V))

for a particular sigma model metricgij . The bulkpotential is given by

(37)V =WiWi −W2

as a function of the superpotentialW . On shell thebosonic Lagrangian (36) is supplemented with theboundary term

(38)Sbound= − 3

2κ25

∫d5x (δx5 − δx5−R)

(√−g4W).

In the case of a single scalar field in the bulk thesuperpotential reads

(39)W = ξeαφ,

where ξ is a scale. The corresponding solutions inconformal coordinates are given by [5]

(40)a =(u

u0

)1/(4α2−1)

.

Supergravity implies thatα = 1/√

3, −1/√

12. Forthe latter we find thatβ = −3/2.

Fortunately, the bulk singularity is forbidden bythe off-shell formulation of supergravity in singular

Ph. Brax, A.C. Davis / Physics Letters B 513 (2001) 156–162 161

spaces. The off-shell theory depends on two newfields. There is a supersymmetry singletG and a fourformAµνρσ . One also introduces a modification of thebulk action by replacingg → G and adding a directcoupling

(41)SA = 1

4!κ25

∫d5x εµνρστAµνρσ ∂τG.

The boundary action is taken as

Sbound= − 1

κ25

∫d5x (δx5 − δx5−R)

(42)×(√−g4

3

2W + 2g

4! εµνρσAµνρσ

).

The supersymmetry singletG satisfies a first orderconstraint

(43)∂x5G= 2g(δx5 − δx5−R),

where the left-hand side is nothing but theAµνρσ

charge associated with the boundary branes. Theconstraint (43) leads to the topological obstruction ofsingularities in the bulk. Indeed, from

(44)∫dx5 ∂x5G= 0

due to the compactness of the fifth dimension wededuce that the total charge in the extra dimensionmuch vanish. This is the equivalent to Gauss’ law,or the tadpole cancellation mechanism in M-theoryand string theory. If one were to have a singularityin the bulk, the total charge would not vanish unlessthe singularity carries a charge, i.e., the extreme casewhere the singularity sits at the second brane. In allother cases the topological obstruction requires thatthe singularity be screened off by the second brane.

The same mechanism is at play when supersymme-try is broken on the brane by detuning one of the ten-sions. In that case the boundary Lagrangian becomes

Sbound= − 1

κ25

∫d5x δx5

( √−g43T

2W

(45)+ 2g

4! εµνρσAµνρσ

)on the non-supersymmetricbrane. The supersymmetrybreaking parameter isT = 1. The Lagrangian on thesupersymmetric brane is not modified. As the branecharge is not modified, the first order constraint (43)

remains leading, to the same topological obstructionas in the supersymmetric case. So supergravity insingular spaces leads to a natural resolution of thesingularity puzzle.

However, it may appear that the tension on thesecond brane (42) has been fine-tuned to the oppositevalue of the tension of the brane-world. As such thiswould be a phenomenon akin to the one presentedin [3] where the ghost brane and the brane-worldhave the same tension. In the case of supergravity insingular spaces the mechanism is more subtle. Indeed,Gauss’s law implies that the total charge vanishes.Hence, by supersymmetry, this leads to the vanishingof the total tension. Now, when supersymmetry isbroken on the brane-world by modifying the tensionof the brane-world, we lose the exact cancellationbetween the brane tensions. Nevertheless, the totalcharge still has to vanish, implying that the would-besingularity is screened off by the second brane.

5. Conclusions

In this Letter we have analysed the naked singulari-ties inherent in self-tuned branes or the supergravity insingular spaces. These theories have been put forwardas possible five-dimensional explanations to the cos-mological constant problem. We have shown that, forsupergravity in singular spaces, the singularity prob-lem resolves itself. This is due to the existence of a to-pological obstruction, requiring by Gauss’s law thatthe total charge vanishes. Thus in this theory the sin-gularity must lie beyond the second brane, unless thesingularity itself carries a charge, in which case it sitsat the second brane. Thus there is a natural resolutionto the singularity puzzle in this theory.

For the self-tuned brane there is no such natural res-olution. We have analysed the behaviour of the sin-gularity both classically and quantum mechanically.Classically the singularity attracts massless particles,in a finite amount of conformal time, this being dueto the behaviour of the scale factor close to the singu-larity. Quantum mechanically we have shown that thesingularity is arepulson, reflecting all incoming radia-tion. However, it is not wave-regular, so that time evo-lution of wave packets is not uniquely defined in thevicinity of the singularity.

162 Ph. Brax, A.C. Davis / Physics Letters B 513 (2001) 156–162

Our results suggest that the theory of supergravityin singular spaces is a well-defined theory cosmologi-cally. We have previously shown that this theory leadsto a natural cosmological evolution of the universe,with a late stage of acceleration and cosmological con-stant consistent with experiment [6]. Thus this modeldeserves further investigation [16].

Acknowledgement

This work was supported in part by PPARC. Ph.B.thanks DAMTP and A.C.D. thanks CERN for hospi-tality while this work was in progress.

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