aa r X i v : . [ h e p - t h ] S e p DAMTP-2007-73
Bouncing Negative-Tension Branes
Jean-Luc Lehners and Neil Turok
DAMTP, CMS, Wilberforce Road, CB3 0WA, Cambridge, UK.
Abstract
Braneworlds, understood here as double domain wall spacetimes, can be de-scribed in terms of a linear harmonic function, with kinks at the locations of theboundary branes. In a dynamical setting, there is therefore the risk that the bound-ary brane of negative tension, at whose location the value of the harmonic functionis always lowest, can encounter a zero of this harmonic function, corresponding tothe formation of a singularity. We show that for certain types of brane-bound mat-ter this singularity can be avoided, and the negative-tension brane can shield thebulk spacetime from the singularity by bouncing back smoothly before reaching thesingularity. In our analysis we compare the 5- and 4-dimensional descriptions of thisphenomenon in order to determine the validity of the moduli space approximation.
E-mail: [email protected], [email protected].
Introduction
Recently, a solution to the classical equations of motion of heterotic M-theory was found,which describes a “non-singular” collision of the two boundary branes [1]. By non-singularwe mean here that the volume of the internal Calabi-Yau manifold, as well as the scalefactors on the branes remain finite and non-zero at the collision, with only the orbifolddimension shrinking to a point. Since domain wall solutions are usually described interms of a linear harmonic function, one might however expect on general grounds that ina time-dependent context a zero of the harmonic function and thus a spacetime singularitymight be encountered at some other point in the evolution. This is indeed the case. Thezero of the harmonic function in fact corresponds to a timelike naked singularity, whichthe negative-tension brane runs into in the absence of matter on the branes. This is theinstability described by Gibbons et al. [2] and by Chen et al. [3] .However, in the presence of a small amount of certain types of brane-bound matter,the negative-tension brane bounces off the naked singularity without touching it. Thisbehaviour is only possible due to the peculiar properties of gravity on a brane of neg-ative tension, and in a sense one can say that in these cases the naked singularity actsrepulsively with respect to the negative-tension brane. Thus, and perhaps paradoxically,the negative-tension boundary brane can have a stabilising effect by shielding the bulkspacetime from the naked singularity that corresponds to the zero of the harmonic func-tion (note that because the negative-tension brane corresponds to a trough-like kink, it isalways the negative-tension brane, rather than the positive-tension one, which will be theclosest to a zero of the harmonic function). It was shown in [5] that from a 4d effectivepoint of view, the bounce of the negative-tension brane corresponds to a reflection of thesolution trajectory off a boundary of moduli space. This reflection has the consequenceof converting entropy perturbations into curvature perturbations [6], and is thus rathersignificant in the context of ekpyrotic [7] or cyclic [8] cosmological models. In the presentpaper we study the conditions for such a bounce to occur in greater generality. Whatwe find is that a certain inequality, involving the trace of the brane matter stress-energytensor and its coupling to the scalar supporting the domain walls, has to be satisfied inorder for a bounce to be possible.We will study the conditions for a bounce both in 5 dimensions and using the 4d It was shown in [4] that static Hoˇrava-Witten braneworlds are stable subject to perturbations offinite energy. However, the time-dependent configurations described in [3] and [1] differ from the staticconfiguration by a homogeneous, infinite-energy perturbation.
We will consider scalar-gravity theories with an exponential scalar potential. The actionis given by S = Z d √− g [ R −
12 ( ∂φ ) − α (3 β − e βφ ]+12 α Z d, y = − √− g e βφ − α Z d, y =+1 √− g e βφ , (2.1)where α is a positive constant that can be adjusted by a shift in the scalar φ (we willchoose a convenient value later on) and β determines the self-coupling of φ. Theories ofthis type are well-motivated in a supergravity context, where they can arise after fluxcompactification `a la
Scherk-Schwarz, see for example [9]. Typically, the domain wallaction is given by a worldvolume-weighted superpotential ∓ Z d, y = ± √− g W ( φ ) , (2.2)where here W ( φ ) = 12 αe βφ . This superpotential is then related to the potential V ( φ ) =6 α (3 β − e βφ by the usual supergravity relationship V = 18 [( ∂W∂φ ) − W ] , (2.3)see [10] and the appendix of [11] for more details. The case β = − e φ then parameterisesthe volume of the internal Calabi-Yau manifold.The static vacuum of the theory above is given by a domain wall spacetime of the3 Figure 1:
The harmonic function h ( y ), where y is the coordinate on a S / Z orbifold. In theabsence of a negative-tension brane at y = − , there would have been a singularity at y = S. form d s = h / (6 β − ( y ) (cid:2) B ( − d τ + d ~x ) + A d y (cid:3) ,e φ = A − /β h − β/ (6 β − ( y ) ,h ( y ) = α (6 β − y + D, (2.4)where A , B and D are arbitrary constants and h ( y ) is a linear harmonic function. The y coordinate is taken to span the orbifold S / Z with fixed points at y = ±
1. In the‘upstairs’ picture of the solution, obtained by Z -reflecting the solution across the branes,there is a downward-pointing kink at y = − y = +1.These ensure the junction conditions are satisfied, with the negative-tension brane beinglocated at y = − y = +1. The coordinate system usedabove is only a good coordinate system when β > , (2.5)and we will restrict our analysis to this range of β (as discussed recently in [16], for certainphysical properties there are qualitative differences when 0 ≤ β ≤ ).The Ricci scalar is proportional to h − β / (6 β − and thus the spacetime is singular at h ( y ) = 0. If we had only a positive-tension brane, with a roof-type kink, this singularity4ould be at a finite proper distance from the brane, and the spacetime would thereforehave a naked singularity. Usually, one avoids this problem by cutting the spacetimeoff with a negative-tension brane placed in between the positive-tension brane and thesingularity, thereby rendering the spacetime well-behaved, as we have already anticipatedby including two brane actions of opposite tension in the action (2.1), see also Figure 1.In a time-dependent context however, where the slope and the height of the harmonicfunction can vary, there is still the risk that the harmonic function can become zero atthe location of the negative-tension brane, thus causing a spacetime singularity to form[2, 3]. In the next section we will see that, in the presence of certain types of brane-boundmatter, this singularity can be avoided, with the negative-tension brane bouncing backbefore it reaches the singularity. In general, we add the following matter action at the location of the negative-tensionbrane (at y = − i.e. we add to equation (2.1) the term:+ Z d,y = − L ( g, φ, ... ) , (3.1)where the dots represent the matter contribution and we are allowing for a coupling tothe scalar φ . The junction conditions, which we are only writing out here for the negative-tension brane, read (in this section ′ ≡ ∂∂y and ˙ ≡ ∂∂t ) a ′ = αe n + βφ + 16 e n T | y = − (3.2) n ′ = αe n + βφ − e n T + 16 e n T ii | y = − (3.3) φ ′ = − αβe n + βφ + 12 e n T φ | y = − , (3.4)where we have defined T µν ≡ − √− g δ L δg µν (3.5) T φ ≡ − √− g δ L δφ , (3.6)with µ a brane worldvolume index. Since the brane as well as the brane-bound matterare kept at the fixed coordinate position y = − , we have T µy = 0 .
5e are only interested in whether or not the negative-tension brane will bounce offthe singularity, even if the bulk is perturbed in the vicinity of this bounce. Therefore wewill choose a general metric and scalar field ansatz, which however respects cosmologicalsymmetry on the brane worldvolumes, so that, on the branes, we have spatial homogeneityand isotropy: d s = e n ( t,y ) ( − d t + d y ) + e a ( t,y ) d ~x (3.7) e φ = e φ ( t,y ) . (3.8)With this metric ansatz we need T i = 0 for the 0 i Einstein equation to be satisfied.As a minimal requirement for a bounce to occur, there should be a solution in which thenegative-tension brane is momentarily stationary ( i.e. for which all first time derivativesare zero at the location of the negative-tension brane), and in which the second timederivative of the scale factor on the negative-tension brane is positive. The yy bulkEinstein equation, which is an equation for the acceleration of the scale factor a , is givenby 3¨ a − a ˙ n + 14 ˙ φ + 6 ˙ a = 3 a ′ + 3 a ′ n ′ − φ ′ + 9 α (6 β − e n +2 βφ . (3.9)We can set first time derivatives to zero, since we are only interested here in the momentof the bounce. Apart from the ty Einstein equation (which is trivially satisfied at y = − y , and so we can evaluate it at the location ofthe negative-tension brane at the moment of the putative bounce by substituting in thejunction conditions (3.2)-(3.4):3¨ a = α e n + βφ ( T µµ + 3 β T φ ) − e n
48 [4( T ) − T T ii + 3( T φ ) ] | y = − (3.10)The first line is proportional to α , and would therefore flip sign on the positive-tensionbrane (where there would be additional first time-derivative terms involved). The first linealso involves the trace of the matter stress-energy tensor. The second line is proportionalto the matter density squared, and can thus be regarded as small compared to the firstline. The second line generally gives a negative contribution (it certainly does so whenthe strong energy condition is satisfied).If we want to have a bounce on the negative-tension brane, there must be a positivecontribution to ¨ a from the first line in (3.10), i.e. a necessary condition (but not sufficient6n general) is that T µµ + 3 β T φ > . (3.11)This condition is not particularly difficult to satisfy; we will give a few examples (andcounter-examples) in the next section. If equation (3.11) is satisfied, then one also has tocheck that this contribution is dominant over the second line in (3.10), which it is if thematter density is sufficiently small. And one would of course have to extend the solutionto the rest of spacetime, which we simply assume here to be feasible. Scalar Field
Using the above equations, one can see that for a scalar matter Lagrangian L = −√− g
12 ( ∂σ ) C ( φ ) , (4.1)where we allow for a coupling C ( φ ) and where we take σ to depend only on time (becauseof the assumed cosmological symmetry) , we get a positive contribution to (3.10) when C − βC ,φ > . (4.2)Thus for a scalar field that doesn’t couple to φ , i.e. for which C = 1 , we can expect abounce; however there will also be corrections to the geometry. Scalars of this latter typeare present in heterotic M-theory [15]. We will discuss the heterotic M-theory examplesin more detail in section 6. Gauge Field
A vector gauge field localised on the brane is represented by the Lagrangian L = −√− gC ( φ ) F µν F µν . (4.3)Here we assume the gauge field to be abelian, and we use the usual electric-magneticdecomposition F i = E i F ij = ǫ ijk B k . (4.4)This leads to a stress-energy tensor T = − g ( E + B ) C ( φ ) (4.5) T i = − ǫ ijk E j B k C ( φ ) (4.6) T ij = [ − E i E j − B i B j + g ij ( E + B )] C ( φ ) , (4.7)7here we have denoted B = ( B i B i ) / . We can immediately see that the stress-energytensor is traceless, T µµ = 0 . (4.8)We also have T φ = C ,φ ( − E + 2 B ) . (4.9)The oi Einstein equation implies that T i , and thus the Poynting vector, has to be zero.This will be the case if we have an electric or a magnetic field only. Thus, from (3.11),we can expect a bounce if βC ,φ < B i = 0 (4.10)or if βC ,φ > E i = 0 . (4.11)On the other hand, it is easy to see that radiation alone, for which the Poynting vectoris zero on average, does not give rise to a bounce, since then h E i = h B i . (4.12)In that case the condition (3.11) cannot be fulfilled, as we now have T µµ + 3 β T φ = 0.However, radiation also doesn’t lead to a collapse; to first order in the matter density itsimply has no effect at all on whether we have a bounce or not. It is only at second orderin the energy density that radiation contributes towards a collapse, as can be seen fromequation (3.10). Perfect Fluid and Cosmological Constant
A perfect fluid with energy density ρ can be described by the Lagrangian L = −√− gρC ( φ ) , (4.13)which leads to the stress-energy tensor [17] T = − g ρC ( φ ) (4.14) T ij = g ij pC ( φ ) (4.15) In order to perform the averaging, we are assuming here that C ,φ varies slowly. T φ = ρC ,φ , (4.16)where p denotes the fluid’s pressure. With an equation of state p = wρ and ρ >
0, we geta bounce if βC ,φ > − w C. (4.17)Note that due to the coupling to the scalar φ , radiation should not be represented asa perfect fluid with w = , but rather as a gauge field, as above. In fact, for that samereason, it is doubtful to what extent the perfect fluid effective description is accurate ingeneral, except in the case of a cosmological constant, which we write out explicitly here.For a brane-localised cosmological constant Λ, we would consider L = −√− g C ( φ ) . (4.18)Then T µν = − Λ g µν C (4.19)and the condition (3.11) is satisfied forΛ( βC ,φ − C ) > . (4.20)Thus, for a positive cosmological constant Λ > e cφ with βc > . (4.21)If we have a negative cosmological constant, we can have a bounce if the coupling is e cφ with βc < . (4.22)Note that when C = e βφ , the addition of a cosmological constant corresponds to a de-tuning of the brane tensions, since it effectively changes the value of α in the brane actionat y = − For many reasons, not least because of our lack of intuition about higher-dimensional set-tings and in order to make contact with what we can observe at present, it is useful to havea 4-dimensional effective description of higher-dimensional physics. An obvious question9owever is how much of the higher-dimensional dynamics a 4d effective description cancapture. We will address this question by looking at the 4d moduli space approximationfor the examples presented in the previous section. The derivation of the moduli spaceaction in this section will be a generalisation to arbitrary β of the derivation in [5], whereit was performed for the case β = − . To implement the moduli space approximation, we simply promote the moduli of thestatic solution (2.4) to arbitrary functions of the brane conformal time τ , yielding theansatz: d s = h / (6 β − ( τ, y ) (cid:2) B ( τ ) ( − d τ + d ~x ) + A ( τ ) d y (cid:3) ,e φ = A − /β ( τ ) h − β/ (6 β − ( τ, y ) ,h ( τ, y ) = α (6 β − y + D ( τ ) , − ≤ y ≤ +1 . (5.1)This ansatz satisfies the τ y Einstein equation identically, which is important, since oth-erwise the τ y equation would act as a constraint [18]. Having defined the time-dependentmoduli, we would now like to derive the action summarising their equations of motion.This is achieved by simply plugging the ansatz (5.1) into the original action (2.1), yieldingthe result (where we use the notation ˙ ≡ ∂/∂τ ) S mod = 6 Z d AB I β − (cid:2) β (cid:16) ˙ AA (cid:17) − (cid:16) ˙ BB (cid:17) − ˙ A ˙ BAB + 3 β − β − I − β β − I β − ˙ D − β − I − β β − ˙ B ˙ DI β − B (cid:3) , (5.2)where we have defined I n = Z − dy h n = 1( n + 1) α (6 β −
1) [( D + α (6 β − ( n +1) − ( D − α (6 β − ( n +1) ] . (5.3)This action can be greatly simplified by introducing the field redefinitions a ≡ A B I β − , (5.4) e r β β ψ ≡ A ( I β − ) β / (3 β +1) , (5.5)(6 β − χ ≡ − Z d D (cid:2) (3 β − I − β β − I β − + β +4 ( I − β β − ) (cid:3) / I β − . (5.6) Note that the relationship between the coordinates ( τ, x, y ) used in this section and the coordinates( t, x, y ) used in the previous section is in general rather complicated. We will not need the correspondingcoordinate transformations in this paper. a has the interpretation of being roughly the four-dimensional scale factor,whereas ψ and χ are four-dimensional scalars. The definition (5.6) can be rewritten asstating that p β + 1 d χ = − d D ( D + α (6 β − (3 β − / (6 β − ( D − α (6 β − (3 β − / (6 β − I β − . (5.7)This expression can be integrated to yield D = α (6 β − " (1 + e √ β +1 χ ) (6 β − / (3 β +1) + (1 − e √ β +1 χ ) (6 β − / (3 β +1) (1 + e √ β +1 χ ) (6 β − / (3 β +1) − (1 − e √ β +1 χ ) (6 β − / (3 β +1) . (5.8)In terms of a , ψ and χ the moduli space action (5.2) then reduces to the remarkablysimple form 16 S mod = Z d [ − ˙ a + a ( ˙ ψ + ˙ χ )] . (5.9)The minus sign in front of the kinetic term for a is characteristic of gravity, and in factthis is the action for gravity with scale factor a and two minimally coupled scalar fields.Note that all the different 5d theories, with different β , are thus described by the same 4deffective theory to a first approximation. We will see shortly however that the inclusionof brane-bound matter lifts this degeneracy.Useful expressions relating 4d and 5d quantities at the location of the negative-tensionbrane are given by: b − = ( α (6 β + 2)) / (6 β +2) a e − r β β ψ ( − sinh p β + 1 χ ) / (3 β +1) (5.10) e φ − = ( α (6 β + 2)) − β/ (6 β +2) e − β r β β ψ ( − sinh p β + 1 χ ) − β/ (3 β +1) , (5.11)where b − denotes the brane scale factor b − = h / (6 β − ( τ, y = − B ( τ ) . Note that since b − is a positive quantity, the range of χ should be restricted to ( −∞ , . For simplicitywe will set α = 1 / (6 β + 2) in what follows; this can be done by a shift in φ. Also, in thissection we always assume the coupling function C ( φ ) to be of the form C ( φ ) = e cφ . (5.12)In heterotic M-theory ( β = − e φ , this corresponds to the brane-bound matter fields coupling to a power of the volumeof the internal manifold. 11efore continuing, let us present a brief argument which partially explains the sim-plicity of the moduli space action (5.9). This arguments rests on the observation that theoriginal 5d action (2.1) is invariant under the global scaling symmetry g mn → e ǫ g mn (5.13) φ → φ − β ǫ, (5.14)where ǫ is a constant parameter. Under this symmetry, the moduli of the domain wallsolution (5.1) transform as A → e ǫ A (5.15) B → e ǫ B (5.16) D → D. (5.17)This in turn corresponds to the transformations a → e ǫ/ a (5.18) ψ → ψ + s β + 112 β ǫ (5.19) χ → χ. (5.20)Thus we see that this symmetry induces the shift symmetry in ψ. It is also interesting tonote that the absence of an implied shift symmetry in χ is consistent with the fact thatthe range of χ is actually limited, as noted above, and that the absolute value of χ is ameaningful quantity. Scalar Field
For a scalar field σ coupling to the scalar φ via e cφ , with c an arbitrary real number, weget an addition to the effective theory (5.9) of −√− ge cφ g ˙ σ | y = − (5.21)= a e − c/β +1) r β β ψ ( − sinh p β + 1 χ ) ( − βc +2) / (3 β +1) ˙ σ . (5.22)The equation of motion for σ can be solved immediately to give˙ σ = σ a e c/β +1) r β β ψ ( − sinh p β + 1 χ ) (6 βc − / (3 β +1) , (5.23)12here σ is a constant. Also, the equation of motion¨ a a = − ˙ ψ − ˙ χ − σ a e c/β +1) r β β ψ ( − sinh p β + 1 χ ) (6 βc − / (3 β +1) (5.24)together with the constraint (Friedmann equation)˙ a a = ˙ ψ + ˙ χ + σ a e c/β +1) r β β ψ ( − sinh p β + 1 χ ) (6 βc − / (3 β +1) (5.25)lead to a = τ / . (5.26)If we then define a new time variable T ≡ ln τ, (5.27)the remaining equations of motion can be expressed as ψ ,T T + σ V ,ψ = 0 (5.28) χ ,T T + σ V ,χ = 0 , (5.29)or, equivalently, by the action Z d ψ ,T + χ ,T − σ V ( ψ, χ ) . (5.30)The effective potential is given by V = e c/β +1) r β β ψ ( − sinh p β + 1 χ ) (6 βc − / (3 β +1) . (5.31)Therefore, as χ → βc < / . (5.32)Thus the solution trajectory effectively gets reflected off the χ = 0 plane which meansthat the scale factor on the negative-tension brane starts increasing again (see equation(5.10)), i.e. the negative-tension brane bounces. Condition (5.32) is the same as thatderived above from the 5d point of view in section 4. This constraint arises from the time reparameterisation invariance of the action or, equivalently, fromthe 00 Einstein equation. auge Field By adding a vector gauge field with Lagrangian L = −√− ge cφ F µν F µν | y = − , (5.33)we obtain an effective theory described by the action S = Z d [ − ˙ a + a ( ˙ ψ + ˙ χ ) − a e − c/β ) r β β ψ ( − sinh p β + 1 χ ) − βc/ (3 β +1) F µν F µν ] . (5.34)Then we have the constraint˙ a a = ˙ ψ + ˙ χ + a e − c/β ) r β β ψ ( − sinh p β + 1 χ ) − βc/ (3 β +1) (2 E + 2 B ) (5.35)together with the equations of motion¨ a a = − ˙ ψ − ˙ χ (5.36)¨ ψ + 2 ˙ a a ˙ ψ + 1 a ∂∂ψ [ e − c/β ) r β β ψ ( − sinh p β + 1 χ ) − βc/ (3 β +1) ]( − E + B ) = 0(5.37)¨ χ + 2 ˙ a a ˙ χ + 1 a ∂∂χ [ e − c/β ) r β β ψ ( − sinh p β + 1 χ ) − βc/ (3 β +1) ]( − E + B ) = 0 (5.38) ∂ µ [ F µν e − c/β ) r β β ψ ( − sinh p β + 1 χ ) − βc/ (3 β +1) ] = 0 . (5.39)The last equation, supplemented by the Bianchi identity ǫ µνρσ ∂ ν F ρσ = 0 , (5.40)leads to E = E e c/β ) r β β ψ ( − sinh p β + 1 χ ) βc/ (3 β +1) (5.41) B i = B i, (5.42)where E and B i, are constants. The equations of motion for ψ and χ can then berewritten as ¨ ψ + 2 ˙ a a ˙ ψ + 1 a V ,ψ = 0 (5.43)¨ χ + 2 ˙ a a ˙ χ + 1 a V ,χ = 0 (5.44)14ith the effective potential V = E e c/β ) r β β ψ ( − sinh p β + 1 χ ) βc/ (3 β +1) + B e − c/β ) r β β ψ ( − sinh p β + 1 χ ) − βc/ (3 β +1) . (5.45)Thus we can see that near χ = 0 the effective potential blows up and leads to a bounceof the negative-tension brane if we either have an electric field and βc < , (5.46)or if we have a magnetic field and βc > . (5.47)This is in agreement with the 5d description of section 4. Also, if we consider radiation,for which h E i = h B i , (5.48)it is immediately apparent from equations (5.36)-(5.38) that it does not lead to a bounce.This is again consistent with the 5d results derived earlier. Cosmological Constant
We can repeat the above analysis in the case of a brane-localised cosmological constantΛ , also coupling to the scalar φ . In that case the effective action receives an additionalcontribution of −√− ge cφ | y = − (5.49)= − a e − c/β +2) r β β ψ ( − sinh p β + 1 χ ) ( − βc +4) / (3 β +1) . (5.50)Therefore, the effective potential is V = Λ e − c/β +2) r β β ψ ( − sinh p β + 1 χ ) ( − βc +4) / (3 β +1) (5.51)and as χ → , we get a bounce if βc > / > . (5.52)This is exactly the same requirement as that obtained from the 5d point of view for apositive cosmological constant.However, the case of a negative cosmological constant cannot be reproduced withinthe 4d effective theory, as the effective potential is negative in that case.15 Heterotic M-Theory Examples
Heterotic M-theory corresponds to the special case β = − , with the scalar φ parame-terising the volume of the internal Calabi-Yau manifold [14]. It is in this theory that thecolliding branes solution [1], which was briefly discussed in the introduction and whichmotivated the present work, was derived. The solution was described in a coordinate sys-tem in which the bulk is static and the branes are moving. The boundary conditions usedcorrespond to requiring the brane scale factors and the Calabi-Yau volume to be non-zeroand finite at the collision of the branes. This turns out to be equivalent to imposing therelationship [1] φ = 6 a. (6.1)This condition relates the volume of the Calabi-Yau to the brane scale factors, whilereducing the number of independent fields to two. This last feature enables one to derivea Birkhoff-like theorem , which determines the bulk metric to be given by a one-parametertime-independent family of metrics (the parameter being the relative rapidity of the branesat the collision), with the branes moving in this background geometry according to theirjunction conditions. It is easy to see from the junction conditions (3.2)-(3.4) that we cankeep the requirement that φ = 6 a, and thus the Birkhoff-like theorem mentioned above,only if T = 12 T φ . (6.2)Thus we can see that in general a very specific coupling C ( φ ) to the Calabi-Yau volumescalar is required if we want the bulk spacetime to remain unaltered by the presence ofbrane-bound matter (the brane trajectories will of course be modified in any case).For a brane-bound scalar, it is straightforward to see that the bulk geometry is unal-tered only if the coupling is C = e φ . (6.3)As shown in section 4, there will also be a bounce in this case, and the entire evolutioncan be described exactly, since the bulk spacetime is given by the solution describedin [1]. From the moduli space point of view, we can note that the effective potential(5.31) is independent of ψ only for C = e φ , which coincides with the condition for thebulk geometry to be unaltered. This can be understood by the fact that, if the effective For the case of general β, a similar Birkhoff-like theorem can be derived if one imposes φ = − βa. The discussion in the present section can be generalised in a straightforward, but unilluminating way tohaving arbitrary β. ψ , the scalar field space trajectory reflects off the effectivepotential with the same final angle as the incident angle, in a smoothed-out version of a“brick wall” reflection at χ = 0 , and therefore the background trajectory is unchangedexcept for this symmetric rounding off of the trajectory near the bounce of the negative-tension brane. Thus, for scalar field matter, the 4d and 5d points of view are in perfectagreement. This can be traced back to the fact that we are simply extending the modulispace by one dimension, by adding an extra kinetic term, and therefore the moduli spacedescription should remain a good approximation.Note that the scalars arising from the dimensional reduction of the E gauge fieldsin heterotic M-theory do not couple to the Calabi-Yau volume, i.e. they have C = 1[15]. Scalars of this type also make the negative-tension brane bounce. However, the bulkgeometry will be altered in this case, which is why it might be of interest to calculate theresulting deformed geometry.For gauge fields, condition (6.2) shows that the bulk is unaltered only if − ( E + B ) C = ( − E + B ) C ,φ . (6.4)This can be satisfied either if we have an electric field only ( B = 0) with the coupling C = e φ (6.5)or if we only have a magnetic field ( E = 0) and the coupling C = e − φ . (6.6)However, in both cases, the effective potential (5.45) in the moduli space description isindependent of ψ only if C = 1 . While the moduli space approximation correctly predictswhether or not a bounce occurs, the detailed trajectory followed in this description is notperfectly symmetric about the bounce (when the coupling is such that the bulk remainsunaltered), and hence not a perfect rendition of the 5d solution.In fact, the E gauge fields in heterotic M-theory couple with C = e φ [15]. Theirelectric component therefore contributes to a bounce, while also leaving the bulk geom-etry unaltered, while their magnetic component rather contributes to a crunch (and adeformation of the bulk geometry).Again by inspection of (6.2), it is easy to see that a brane-bound cosmological constantdoes not perturb the bulk geometry if its coupling is given by C = e − φ . In this case, wesimply have a de-tuning of the brane tension. This de-tuning leads to a bounce if the17osmological constant is positive, whereas it leads to a crunch if it is negative. Note thatthe moduli space description yields a potential (5.51) that is independent of ψ only when C = e φ , which is in disagreement with the 5d description. In a dynamical braneworld setting, the negative-tension boundary brane can encounter azero of the harmonic function corresponding to the formation of a singularity. However, wehave shown that this catastrophic encounter is avoided in the presence of a broad range ofbrane-bound matter types and couplings to the scalar field supporting the domain walls,which make the negative-tension brane bounce off the naked singularity . This leads usto the rather surprising conclusion that negative-tension branes can stabilise braneworlds.We have analysed the bounce of the negative-tension brane from two points of view:firstly, we have looked at the 5d equations of motion and junction conditions in thevicinity of the bounce. And secondly, we have analysed the analogous situation usingthe moduli space approximation. For scalar fields, the two descriptions are in perfectagreement. This is because adding a kinetic term is perfectly suited to the spirit of themoduli space approximation. For gauge fields and for a positive cosmological constant,the moduli space approach correctly reproduces the 5d results for the bounce. However,when the conditions are fulfilled for the 5d bulk to remain unaltered and we hence knowthat the 4d trajectory should be perfectly symmetric about the bounce, the 4d effectivetheory does not reproduce this behaviour. And in the case of a negative cosmologicalconstant, the moduli space approach completely disagrees with the 5d results. It seemsclear that in case of a disagreement, we should rather trust the 5d results. In fact, ourresults indicate that in the case of a brane-bound gauge field or a cosmological constant,the approximations used in deriving the moduli space action are not really valid. In thesecases, there are non-flat directions in configuration space which are easily accessible tothe system under study, and which are not described by the moduli space approximation.Thus, even though the moduli space description can give qualitatively correct results indescribing the effects of a gauge field or a positive cosmological constant, the detailedquantitative analysis can be rather misleading, and one should revert to a 5d description.The types of brane-bound matter that are naturally present in heterotic M-theory Thus, we could say that we have a bang if no observer is there to hear it, but no sound in the presenceof the right kind of observer! e φ coupling. What we found is that for this specific coupling, electric fields contributetowards a bounce, while radiation has no effect and magnetic fields rather contributeto a crunch. The scalars contribute towards a bounce, and probably represent the bestcandidates for stabilising the heterotic M-theory braneworld.Finally, we would like to point out that it seems likely that additional brane-boundmatter will be produced by quantum effects at the bounce of the negative-tension brane,and it would be interesting to determine the properties of these new contributions. Acknowledgements
The authors would like to thank Gary Gibbons, Paul McFadden, Paul Steinhardt andKelly Stelle for useful discussions. The authors are supported by PPARC and the Centrefor Theoretical Cosmology in Cambridge.
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