DAMTP-1998-85Ination, Singular Instantons and Eleven Dimensional CosmologyS.W. Hawking

and Harvey S. Really

DAMTP Silver Street

Cambridge, CB3 9EW, UK (December 10, 2001)

Abstract

We investigate cosmological solutionsofeleven dimensional supergravity com

pacti ed on a squashed seven manifold. The e ectiveaction for the fourdimensional theory cont

ainsscalar eldsdescribing the sizeand squashing ofthe compactifying space.The potentialfor these

eldsconsists ofa sumofexponentialterms. Atearly timesonly one suchtermis expected todominate.

The condition for an exponentialpotentialtoadmit inationary solutionsis derived and it is sh

own thatination is notpossible inourmodel.The criterion for an exponentialpotentialtoadmit

a Hawking-Turok instanton is alsoderived.It is shown thatthe instanton remainssingularineleve

n dimensions.

Typeset using REVT E X

email: S.W.Hawking@damtp.ca

m.ac.uk

y

email: H.S.Reall@damtp.cam.ac.uk

1I. INTRODUCTION

Until recently it was thought that slow-roll ination always gives rise to a at universe. This assumption was proved incorrect in [1,2] (building on the earlier work of [3] and [4] on bubble nucleation and ¡®old¡¯ ination) where it was demonstrated that an open universe can arise after quantum tunneling of a scalar eld initially trapped in a false vacuum. (One can also obtain open ination in two eld models [5,6]). However such models of open ination appear rather contrived owing to the special form that the scalar potential must be assumed to take. They also do not address the problem of the initial conditions for the universe i.e. no explanation is given of how the scalar eld became trapped in the false vacuum. These two objections were confronted in [7] within the framework of the ¡®No Boundary Proposal¡¯ [8]. It was described there how an open universe could be created without assuming any special form for the potential. The approach was to construct an instanton (i.e. a solution to the Euclidean eld equations) and analytically continue to Lorentzian signature. The novel feature of the instanton is that it is singular although the singularity is su ciently mild for the instanton to possess a nite action. Several objections have been raised against the use of such instantons, the most serious of which is Vilenkin¡¯s argument [9] that if such instantons are allowed then at space should be unstable to the nucleation of singular bubbles. Another objection is that the singularity can be viewed as a boundary of the instanton (there is a nite contribution to the action from the boundary [9]) which is unacceptable according to the no boundary proposal.

There have been three di erent approaches to dealing with the problems raised by a singular instanton. The rst is to regularize the singularity with matter in the form of a membrane [10,11]. An alternative approach [12] is to analytically continuethe instanton across a deformed surface that does not includethe singularity. The problem with this is that the surface does not have vanishing second fundamental form which means that one obtains a region of spacetime which does not have purely Lorentzian signature. It was pointed out that this region is not in the open universe so it may not have observable consequences. The third approach, due to Garriga [13], is to construct a four dimensional singular instanton from a higher dimensional non-singular one. This approach is particularly appealing because M-theory is elevendimensional. Garriga gives a non-singular ve dimensional instanton that reduces to Vilenkin¡¯s in four dimensions but with a cut-o to the scale of bubble nucleation that makes the decay rate of at space unobservably small. He also gives a ve dimensional solution with cosmological constant that reduces to a four dimensional instanton of HawkingTurok type. (Garriga¡¯s ve dimensional instantons are just Euclidean Schwarzschild and the ve sphere respectively). One purpose of this paper is to examine whether it is possible to obtain Hawking-Turok instantons in four dimensions from non-singular instantons of eleven dimensional supergravity, the low energy limit of M-theory.

Our second aim is to investigate whether solutions of eleven dimensional supergravity corresponding to four dimensional inating universes exist. Since ination is now widely accepted as the standard explanation of several cosmological

arXiv:hep-th/9807100v2 4 Aug 1998

problems (see e.g. [14]), one would expect the existence of inationary solutions of M-theory if it is indeed the correct theory of everything. However compacti cations of D = 11 supergravity usually give a negativecosmological constant (see [15]) which is precisely the opposite of what we need for ination. The reason for this is that if the compactifying space has positive curvature then

2the eld equations imply that our space has negative curvature. This suggests that a way around the problem may be to look for solutions with the seven dimensional compactifying space M7negatively curved at early times but positively curved at late times. We do this by taking M7to be a coset space and squashing it (the meaning of squashing is explained below), treating the squashing parameters as dynamical scalar elds.

Upon reductionto four dimensionswe obtain a modelwith scalar elds evolvingaccording to a potential consisting of a sum of exponential terms. At early times only one term in the sum is expected to be signi cant. Cosmological solutionsinvolving scalar elds with exponential potentials have been investigated by severalworkers. Lucchin and Matarrese [16] showed that power-law ination can result fromsuch potentials. This was further investigated by Barrow [17] who gave an exact scaling solution to the equations of motion which was subsequently generalised by Liddle [18]. Halliwell [19] has conducted a phaseplane analysis of the equations of motion resulting from an exponential potential. Wetterich has derived scaling solutions for cosmologies with the scalar eld coupled to other matter [20]. For the single scalar case we have found a rst integral of the equations of motion and givean exact expression for the number of inationary efoldings. It is found that a signi cant inationary period only results from solutions that approach the scaling solutions at late times. The results are generalized to the multi-scalar case. We have analysed the behaviour of scalars with an exponential potential near the singularity of the instanton and give a criterion for the singularity to be integrable. (This was discussed in [21] but the analysis was incomplete).

Applying the results on exponentialpotentials to our model from elevendimensions yields the disappointing result that the potential is too steep for ination to occur. We nd that unlike in Garriga¡¯s models the instanton is singular inelevendimensions. The reason for this is that Garriga¡¯s potential comes from a ve dimensional cosmological constant whereas ours comes from the Ricci scalar of the compactifying space and has too steep a dependence on the scalar eld that measures the size of the internal space (i.e. its ¡®breathing¡¯ mode). It is this samedependence that rules out inationary behaviour which leads us to speculate that if one could x the size of the internal space then a solution with more appealing properties might be found.7As this paper was nearing completion we received a paper by Bremer et al [22] which has some overlap with our work. They also consider cosmological solutions with dynamical squashing in various dimensions. Their S7example is not the same as ours: they obtain squashed metrics on Sby viewing it as a U(1) bundle over CP37and squashing corresponds to varying the length of the U(1) bres whereas wetreat Sas a S3bundle over S43and squashing corresponds to varying the size of the S bres. Our methods are applicable to any squashed coset space (although we always use the Freund-Rubin ansatz [23]). Integrability of the instanton singularity is not discussed in [22] (indeed the examples discussed there all appear to be non-integrable) and neither is the condition for ination. (In the conclusions section of [22] it is stated that the instanton solutions can be continued to give open inationary universes. This is not the case: the potentials are too steep to yield a

signi cant inationary period).

3II. ELEVEN DIMENSIONAL SUPERGRAVITY

The action for the bosonic sector of D= 11 supergravity is [15

+ 2(12)41 pM1:::M 11^ FM1:::M4FM5:::M8^ AM9:::M11!^g Sboundary: dimensional ones. Upper case Roman letters will be used for eleven dimensional indices, and lower case Greek letters for four dimensional ones. ^ = 8 ^ Gis the eleven dimensional Planck scale. We willM1:::M11is the alternating tensor density. The four form ^ Fuse a positive signature metric and a curvature convention such that a sphere has positive Ricci scalar. ^ Ais related to its three form potentialMNPby ^FMNPQ= 4@[M^ ANPQ]MNPQ(2.2)where square brackets denote antisymmetrization. Sboundaryis a sum of boundary terms which are essential in quantum cosmology:Sboundary= B1+ B2; (2.3)

B2= 16Zd 1x @M q^g F̂MNPQ^ ANPQ : (2.4)

The equations of motion following from the action 2.1 are:

RMN= ^ 26 ^ FMPQRFPQR N1F12 ^PQRS^ FPQRS^gMN ; (2.5)

@Mq^g ^ FMNPQ = 1576 NPQM1:::M8^ FM1:::M4^ FM5:::M8: (2.6)

If F^F^nvanishes on the boundary (where nis the 1-form normal to the boundary) then the action is gauge invariant and the second boundary term is

B2= 124

d11xq^g^

FMNPQ

^ FMNPQ: (2.7)

4III. SQUASHED MANIFOLDS

7Given a Lie group G, the manifolds admitting a transitive action of Gcan be viewedas coset spaces G=H where H is the isotropy subgroup. We are interested in the most general G-invariant metric on such a manifold (i.e. the most general metric forwhich the left action of Gyields a group of isometries). If G=His isotropy irreducible (see [15]) then there is a unique (up to scale) such metric which is actually an Einstein metric. For example if G= SO(8) and H = SO(7) then the unique G-invariant metric on G=H is the round metric on S7. If the coset space is not isotropyirreducible then the general G-invariant metric contains arbitrary parameters. This is what is meant by squashing. An example is G= SO(5) and Hthe SO(3) subgroup such that G=Hhas Stopology. The most general Ginvariant metric contains seven arbitrary parameters and there are two G-invariant Einstein metrics.aIt is discussed in [25] how one can squash a coset space by rescaling the vielbein i.e. e! aea(no summation). A criterion is given for deciding if a particular rescaling will preserve the isometry group of the metric. This is the most general kind of deformation that preserves the G-invariant metric on G=H(see [26] for a review).

IV. DIMENSIONAL REDUCTION WITH DYNAMICAL SQUASHINGOur metric ansatz is

where gand ymmn (y)=2

d^s

a m R=q(x)dx dx + gmn(x;y)dymdyn; (4.1)

ea m(x;y) = eAa(x)ea m(y) (no summation), (4.2)

(y) is the unsquashed metric. The squashing is described by the seven scalar elds A(x). Note that for squashing in the sense described above (i.e. preserving the isometry group) these scalar elds will not be independent.In the following discussion we shall not specify a particular squashed coset for M.The choice is not arbitrary: the eleven dimensional eld equations have to be satis ed. We shall assume that a suitable coset has been found but our conclusions will be independent of the details of the internal space. As an example we shall consider Sas a SO(5)=SO(3) coset with a two parameter familyof metrics (i.e. only two of the Aare independent), one parameter being the size and the other the squashing. This choice does satisfy the D= 11 eld equations.B(x) is calculated by obs erving p gg7ePAa(x)e2B(x)(R+ :::); (4.3) mnP a e(y)ea n 7

7so after integrating over ythe reduced action will be in the Einstein frame p

m

2B(x) = 1aXA(x): (4.4)In the Einstein frame the components of the eleven dimensional Ricci tensor are (see appendix):

R = R + 12 r2a XAa!g 32aXAa ; Aa ; 2 1Xa6=bAa ; Ab ; ; (4.5)

Rab= Rab[M7]ePAdiag 2Aa : (4.6)The four dimensional part has been written with curved indices and the seven dimensional part with tangent space indices for notational clarity.

We shall use the Freund-Rubin ansatz [23] for the four form i.e. ^F (x;y) = F (x) other components vanish. (4.7)

(Some other ans atze for the four form were considered in [22]). One can substitutethese ans atze into the eld equations to obtain equations of motionfor the e ective four dimensional theory. Alternatively one can obtain the same equations by varying the reduced action obtained by substituting into 2.1:S=

Zd

4x pg 1 2 2

R 14

2 + 12 2ea;bX(@APAaa)M

ab

R[M

(@A7b

) +] 148 e3PAa

F F + B; (4.8

)2ab7

2V7. The integration over ym

7

a

Awhere eAis the squashing parameter de ned in appendix B. The two eld equations coming from the internal space give the equations of motion for Aand C. The same equations of motion can be obtained from the four dimensional reduced action.Returning to the general case, the kinetic term can be diagonalised by de ning kk Ajk+1k = 1:::6; (4.10

)

= 3 p147j=1XAj: (4.11)

example: 1= 2= 3= 0; p4:5 4= p5:6 5= p6:7 6= 4A ;

2 (@ 4)2

a

k

The inverse transformation relating Ajq12 7 A

2 (@ )2; (4.13)

a1 k+ p 1 A: (4.14)Since 4, 5and 6to kare not independent we de ne =so that 1+ 12 (@ 5)2+ 12 (@ 6)2= 1

so now the scalar elds Aand Chavebeen replaced by and with diagonal kinetic terms. Note that a scaling of the internal

manifold A(x)!A(x) + C(x) only a ects , which measures its size. In general one must allow the size tovary inorder to satisfythe D

sions later on sAj = 0 @sj1 j j1+6k=j Xq k(k+ 1) 14 21Substituting into the reduced action gives

S=Zd4x pg 1 2 2k=11 2( @ k)2 12W( ; )

148 e p14 F Fw)7

W( k; ) = 12 2ep 143 R[ M7]: (4.16

)The equations of motion following from this action are:

r2 k= @W@ k; (4.18)

r2 = @W@ + p14 48e p 14 F F ; (4.19)

F = pg e

p 14

r2 = @V@ ; (4.22

)where V(

kk

2 F2e p 14

(4.23

)) = 0: (4.20) Note that the nal equation is obtained by varying A. This equation has the unique solution

F; (4.21) for some constant F. If we substitute this solution into 4.19 then we get

; ) = W( ; )+ 1

is the e ective potential that determines the evolution of the eld . Note that we canreplace W by V in the eld equation for kand substitute the solution for F into 4.17 to yield

R = 2 2k X1 2(@ ) (@ k) + 12 (@ )(@ ) + 12 Vg !; (4.24)

7so now V occurs in all of the equations of motion and one can forget about W. For our squashed Sexample, the potential is

V( ; ) = 12 2e3 7p 14 3 2e21p 21+ 12e1 7213e10 2121 + 12F2e p 14 : (4.25)7Plotted as a function of this tends to 1as ! 1. There is a local minimum at = 0 corresponding to the round metric on the S. There is also a local maximum at a

8negative value of corresponding to the squashed Einstein metric on S70 0the F-independent part of the potential vanishes. For > 0

17 S

3

5

,6. The qualitative behaviour as varies depends on the value of . There is a positive constant such that at = , V is a monotonically decreasing function of tending to +1as !and to 0 as !+1. For < (which includes the two Einstein metrics), the asymptotic behaviour is similar but there is a local minimum at somevalue of corresponding to a negative value of V. Hence there exist static solutions of D= 11 supergravity with sitting at this minimum and corresponding to either the round or the squashed Einstein metric. These have been extensivelydiscussed from the point of view of Kaluza-Klein theory [15]. We are interested insolutions with a positive potential at early times in the hope that these may exhibit inationary behaviour. Such solutions start with large and positive, corresponding to a negatively curved metric on S7. One would expect solutions toexist in which rolls down to the local minimum so the solution settles into the Freund-Rubin solution AdS(with a round metric) at late times. Note that this solution appears unstable because can tunnel past the local maximum and roll o to . However Breitenlohner and Freedman [27] have shown how boundary conditions at in nity can stabilise AdS, at least against small perturbations, so onewould expect a similar argument to be valid here.A second example that we have considered involves taking the compactifying space to be S S S3. S3is group manifold so one can squash all three directions independently [15]. Thus this M7can be squashed with all seven Aindependent. The D= 11 eld equations can be satis ed with this M7a. The Ricci scalar of S3withsquashing described by Aand Ais

R= e2A5+ e2A6+ e2A7 12 e(A5A6A7)+ e2(A6A7A5)+ e2(A7A5A6) : (4.26)

Thus in order to get a negatively curved S31the second group of terms must dominate the rst. Note that there is no static Freund-Rubin solution in this case because S S3cannot be given an Einstein metric. Thus the potential V does not have any extrema. S3

V. CONDITION FOR INFLATIONWe shall now seek a solution of the eld equations derived above that describes a four dimensional universe. Spatial homogeneity and isotropy imply that the metric must take the formds2= dt2+ a(t)2ds2 3

2 3

_a a

3k

1

2k 2

+ 12+

V!k;

(5.2

)(5.1) where dsis the line element of a three-space of constant curvature. Substituting this into 4.17 yields the equations

= X_ 2 _a2

a= 23 ak X_ k 2 9+ _ 2V!: (5.3

)kis the sign of the curvature of the spatial sections. Ination is de ned by a>0 so 5.3implies that for ination we need

V >kX_ k 2+ _2: (5.4)

Since the potentials we have obtained consist of a sum of exponential terms with no extrema of at positive values of the potential, one would expect that if ination occurs then it would do so at a large value of the potential where typically only one exponential is signi cant. For simplicity we shall consider a single scalar modelV( ) = V0e : (5.5) The equation of motion for the scalar eld is 3H_ =dV d (5.6)

_ /e1 2 _a a 1 2

(5.7) by1 2d d 1 2d d 2_ 2 + 2 2s3 1 2 2+ V0 + V0, eliminates _ in favour of and neglects the curvature term ( )Now de ne xby = p2V0sinhxto give dxd

=1 2 p 6 cothx 2x: (5.9

)e1 2 ( )= F(y) y

=

0p 6+ )

10

6 +

p6 2

: (5.10

)It is obvious that there is a solution with x= constant when < p 6. This is the solutionobtained previously by Barrow [17] and Liddle [18]. However we can investigate the general solution by using the change of variable y= eto give 2(y p6 2

)instanton to an open universe at a point where the scale factor vanishes. This means that initially it is not a good approximation to neglect the curvature term in the Einstein constraint equation as we have done here. However if there is a signi cant inationary period then the curvature term will rapidly become negligible. So, strictly speaking, our analysis is only applicable after this term has become negligible by which time the above boundary conditions will not hold (since then_ <0 so y>1). However, as we shall show, the condition for a signi cantvaluefor simplicity. Of course our results are exact for at (k= 0) universes. With these boundary conditions, 5.10 becomese1 2 ( 0)= F(y)F(1) : (5.12)

equivalently, sinh2 3 <2. This is satis ed if, and only if,

2 p3: (5.13) Since we are starting with_ = 0 we will always get some ination. How much we get2depends on F(y), the qualitative behaviour of which depends on the magnitude of . There are two cases to consider: i) >6. In the rst case F(y) has zeros at y= 0 and y= y0 p 6+ p 6 and a local maximum at y= 1. For large y, F(y) tends to in nity as a power of y. The solution x= constant corresponds to the second zero of F(y) (butto an2eternally inating universe). If <2 then the second zero of F(y) lies within the range of values of ycorresponding to ination. This implies that the solution will inate all theway to F(y) = 0 i.e. to = . For larger ination will stop before F(y) = 0 is reached. In case ii) the only zero of F(y) is at y = 0. Once again F(y) has a maximum at y = 1, beyond which it decreases monotonically to zero as y!1.We have succeeded in nding a rst integral for the scalar eld equation of motion. This relates and_ implicitly. It does not seem possible to integrate this again to nd an explicitsolution for (t) but this is not necessary in order to calculate the number of inationary efoldings N, de ned bywhere tmax to 2 +

N= 2F(y

)is the (comoving) time at which ination stops. One can now substitute the expression for H(t) in terms of and and then substitute for and in terms of yusing the de nition of yand 5.10. To transform the integral over tinto an integral over yone needs to knowwhich is obtained by di erentiating 5.10 with respect to t.yruns from 1 (whenp 3 (end of ination). This gives

Z2+p3 1 y+ 1 y1! F0! d y; (5.15

)which is in nite if < p2 and otherwise evaluates to

his is small unless is exponentially close to p2. We can conclude that an exponential potential can only give a signi cant inationary period if p 2. Note that the result isin contrast with the result for power law potentials. As mentionN= p6 3(p 6 + )

independent of 00

4

3)+ plog(2 + p6 p6 log 1 + p6 p3( p 2

)!#: (5.16

) = 6 22 (tt0)2; (5.18)

a= a0(tt0) Einstein constraintequation at largetimesif 2

21 2_ 2= 13(tt0)2; (5.20): (5.19) It is clear from these

expression that it is only consistent to neglect the curvature term in the2<2, otherwise these results are restricted to at (k= 0) cosmologies.In case ii) y !1implies x!1. Substituting this into 5.9 and solving gives the following (tt0)1 3 12

2 p6; (5.21

): (5.22) Note that the potential energy density is negligible compared with the kinetic energy densityas t!1(indeed this asymptotic solution may be obtained by simply neglecting V in the eld equations). The curvature term in not negligible in this case so these results are only valid for k= 0.If we include extra scalar elds but still assume that the potential is dominated by a single exponential term Vthen the situation just gets worse because this potentia

lmust now dominate two kinetic terms to yield ination. One can make progress analytically by de ning = . Then the equations of motion for and imply that obeys

+ 3H_ = 0; (5.23)where the Hubble parameter is given by the Einstein constraint equation with twoscalar elds. This equation can be integrated to give_ = Aexp 3ZtH(t0) dt0 ; (5.24)

where Ais a constant. If the scale factor grows su ciently fast then this term will be asymptotically negligible and constant will be a good approximation. Then we can write = + constant. Now de ne

andp 2+ 2 0 0)16 2_ an d otherwise _ /(tt0)

fIf 2

j+

/22

2

2 p 14 21

p

2+

2dependent piece ~ V. Since 3 p14

7 23 p 14

7 . This multiplies a

to concluded that the solution is still inating at large times only when

+ 2

+

<2. By analogy with the single scalar results one would only expect a signi cant inationary period from such solutions.

>6 then an asymptotic solution (for k= 0) can be found in analogy with the single scalar case by neglecting V. The scalar equations of motion can be immediately integrated and the result plugged into the Einstein constraint equation to give the scale factor. The results are similar to the single scalar case. Our resultscan obviously be generalised when there are more than two scalar elds present.

We can now return to our model obtained from eleven dimensional supergravity. The nal (F) term in V is too steep to drive ination so we turn to the

term coming from the (seven dimensional) Ricci scalar. This depends upon the speci c internal manifold that we choose to squash but it is possible to extract the dependence in the general case. To see this, note that the scalars Aall have the same dependence on in 4.14 hence the metric on the internal space depends on only through the conformal factor e. It follows that the dependence of the rst term in V on is given by a factor e

>2, the above work shows that it is not possible to get inationary behaviour driven by a single exponential term in the potential. If inationary solutions are possible then they must arise from a combination of several such terms leading to a less steep region of the potential, for example neara local maximum. However the potential cannot possess a local maximum in and only possesses a local minimum whenination.1

3VI. SINGULAR INSTANTONS

The behaviour of the Hawking-Turok instanton [7] corresponding to an exponential potential can be analysed in a similar manner. The instanton is assumed to possess an O(4) symmetry so its metric can be writtends2= d 2+ b( )2d2

b0b!2= 13 1 2 02V + 1b2; (6.2): (6.1) The Euclidean eld

equations are

b00=13 02+ V b; (6.3)

00+ 3 b0b 0= dVd : (6.4)

Hawking and Turok consider solutions to these equations that are regular at the North pole, where they look locally like four spheres, and singular at the South pole. As the singularity is approached they assume that the scalar kinetic term dominates its potential. We shall investigate this assumption for V = V0e . If the curvature term1 b2is negligible in the Einstein constraint equation then near the singularity we can write

b0b = s1 3 1 2 02V (6.5)

2

d

2V0

dx d

which can be integrat

ed by de ning y= e2x1

2

1 2 ( )= G(y) yy+ p 2 : (6.8)

n= 12

e p 6 tanhx

to give; (6.7

14

Case i) G(y) is a monotonically increasing function so ybecomes large as becomes large. Asymptotically we have

e1 2( ) y 2(p 6 )= ep6 x)x 12 (p 6 ( ): (6.9)

This gives 0e1 2 = q2V0cosh 12 (p 6 ) ( ) sV02 e1 2( p 6 )( ); (6.10)which is a di erential equation that we can solve for to give

e1 2p 6 s3V04 e1 2( p 6 ) ( f ); (6.11)

V /( f )2 p6; (6.12)

so near the singularity the kinetic term will dominate. The behaviour of the scale factor is easily obtained from the Einstein constraint equation:

b b0( f )1 3

0

p 6).

r the singularity to beintegrable,Rd b3

2 e1 2 2 0( f )1: (6.16)

2p 6, y +p 6 6 2 6: (6.15): (6.14) bis a constant that is determined by matching the solution near the South pole to the solution at the North pole. Note that it is consistent to neglect the curvature term near the singularity. (These results agree with those obtained in [21] but it was not pointed out there that they are only valid for <V must converge [7] (one must include the boundary term Bto derive this result). It is easy to see that it does in this case. Case ii) For

Hence near the singularity the potential and kinetic terms behave as follows1

5V 2 2cosh2x0( f )2(6.17)

1 2 02 22( f )2; (6.18)

so now they only di er by a constant factor, which must be taken account of in order to determine the solution for b. Substituting these asymptotic solutions into the Einstein constraint equation and eliminating x0in favour of yields

b b0( f

b3V /( f ) )2 2

2

1 6 2+ 2

7: (6.19) (Hence it is consistent to neglect the curvature term). So now we have

6 2

2; (6.20) but2 <1 so the singularity is not integrable in this case. In the two scalar case, arguments similar to those presented in the previous section show

and then one nds that the exponent is 2 4+6 5 5+6 7 6+21

, so thesum of squares of kcoe cients is80 7>24 7hence the singularity is not integrable in this case either.

VII. SINGULARITY IN ELEVEN DIMENSIONS

The examples provided by Garriga [13] are encouraging evidence in favour of being able to obtain Hawking-Turok instantons in four dimensions by dimensional reduction of higher dimensional non-singular instantons. However Garriga¡¯s cosmological (as opposed to at) example is special because it requires the

presence of a cosmological constant in the higher dimensional theory. This always gives rise to a potential in the dimensionally reduced action that gives both ination and an integrable singularity on the instanton. Eleven1

6dimensional supergravity does not have a cosmological constant which is why we have been considering squashing as an alternative mechanism of generating a positive potential in the four dimensional e ective action. Unfortunately it is easy to see that the instantons of the type that we have considered remain singular even when viewed from within the higher dimensional framework. If one takes the trace of the eleven dimensional eld equations then one obtains

R=^ 2F72 ^PQRS^ FPQRS; (7.1)

remembering that a factor of p V7 p 14

23 F2e2 3: (7.2)Since !at the (four dimensional) singularity independently of the sign of R[M7] (the exponents in V are negative so !rather than +1as considered above) one sees immediately that

curvature singularity.^ R!so

the Hawking-Turok singularity is an eleven dimensionalVIII. DISCUSSION AND CONCLUSION

SThe results of the previous sections are rather disheartening: our aim was to nda non-singular instanton in eleven dimensions that gives rise to a Hawking-Turok instanton in four dimensions and an inationary period after continuing to Lorentzian signature. Instead we have found that our model realizes neither of these objectives. However the stumbling block appears to be the same in both cases. Ination was ruled out because the potential depends too steeply on . The singularity of the eleven dimensional instanton is also due to the dependence on . Note that the eleven dimensional Ricci scalar is independent of kso if some means were found of xing the size of the compactifying space (i.e. keeping constant) then the instanton may become non-singular in eleven dimensions even with k! 1(and hence singular in four dimensions). The problem with keeping xed is in satisfying the eleven dimensional eld equations. One would have to introduce extra degrees of freedom in the metric of the compactifying space, which involves going beyond squashing.7It is conceivable that by modifying our ansatz for the four-form more interesting results might be obtained. The work in this paper is only applicable to cosmologicalsolutions using the Freund-Rubin ansatz [23]. Bremer et al [22] consider solutions with some more general four-form con gurations. However neither of their Sexamples (round or squashed) appear to admit inationary solutions or instantons with integrable singularities. The FreundRubin ansatz is attractive as an explanation of why there are four non-compact spacetime dimensions but leads to a large negative cosmological constant. In [12] it was suggested that this may be balanced by a contribution from supersymmetry breaking. Such symmetry breakingwould also have a dynamical e ect at early times and might generate corrections tothe e ective potential that make ination possible.

1

7APPENDIX A: DERIVATION OF RICCI TENSOR

The non-zero Christo el symbols for the metric 4.1 are given in terms of those for the (D= 11) metric with B= 0 by^ = + B; + B; B; g ^ mn= e2Ba XAa; ea mea n

n =aXAa; em aea n^ m np= m np: (A1)s^ R = R r2B+ 2B; +X A a; ! B; ! g r r 2B +XAa!+

+ 2BB; +X(Aa; B; a + Aa; B; ) aXaAa; Aa; ; (A2

)^ R

Rmn

[M7

mn=Rmn

[M7

e0]

e2B

ia

X

7ea

mea

n"r2Aa

+ Aa;

2 e 2B;

mn+bX

abeAb;

!#

eb

n: (A3

)] is the Ricci tensor of the squashed internal manifold computed treating the scalars Aaas constants. Note that substantial simpli cation occurs when we use the Einstein frame, given by 4.3.

APPENDIX B: THE SQUASHED SEVEN SPHERE

As discussed in section III, S7can be regarded as the coset SO(5)=SO(3). The most general SO(5) invariant metric on Scontains seven parameters. Here we restrict ourselves to a two parameter subset. The particular squashing we use hereis described in more detail in [15].Using letters near the start of the Greek or Roman alphabet to denote tangent space indices, the metric is given in terms of the siebenbein as g= a mwhere= d ; e= 12 sin !i^ i; e= 1A(x)( i+ cos !i): (B1)

A(x)is a scalar eld that measures the amount of squashing. The round S7is given by A= 0. The one forms iand !iare given by i= i+ i; !i= i (B2) where iand ieach satisfythe SU(2) algebra:i1

8

d i= 12 ijk j^ k; d i=12 ijk j^ k: (B3)

The tangent space components of the Ricci tensor are [15]

Rb

where= diag( ; ; ; ; ; ; ); (B4

) = 3(1 12 e2A); = e2A+ 12 e2A: (B5)

Acknowledgements We have enjoyed useful discussions with Steven Gratton and Neil Turok.1

9REFERENCES

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