Toward globally stable compactification in superspace

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29 December 1988

TOWARD GLOBALLY STABLE COMPACTIFICATION IN SUPERSPACE Marek SZYDLOWSKI Astronomical Observatory, Jagiellonian University, ul. Orla 171, Pl-30-244 Cracow, Poland

Received 21 March 1988; revised manuscript received 4 October 1988

The problem of the behaviour of geodesics in minisuperspaceof multidimensional cosmologyis discussed. We show that the compactification mechanism is globallystable if it leads to a Minkowski static microspaceconfiguration.

1. Introduction

Multidimensional theories have recently revived in physics as a natural base for unifying gravity and gauge theories [ 1 ]. It is supposed that the early Universe was a stage on which unified physics had played an important role. On the other hand, the extra dimensions could interact dynamically with the physical space, which changes our imaginations about the early stages of the evolution of the Universe. In the case of homogeneous multidimensional cosmology, for instance, the additional dimensions cause the vanishing of the chaotic behaviour near the singularity in Mixmaster models, which suggests that the early stage of the evolution of the Universe corresponds to Barrow's conception of quiescent cosmology rather than to Misner's chaotic universe [2,3]. Because of the limitation on the variability of physical constants we suppose that the internal space is presently static (or at least slowly variable), and that its size is of the order of the Planck length. The question of how the present state of the Universe evolved from its multidimensional stage is referred to as the problem of compactification (also the term "dynamical dimensional reduction" is used). Many attempts to answer this question lead to the search of a dynamical mechanism of a compactification. One assumes that Einstein's field equations remain valid in ( 1 + 3 + D ) dimensions (where D is the dimension of the internal space), but they are supplemented by different effects such as high-and low-

temperature quantum effects, monopole and superstring effects, etc. [4 ]. The dynamical compactification mechanism is assumed to be correct if it additionally leads to a stable configuration FRW × K, where K is the internal space of constant size [ 5 ]. Although different mechanisms giving F R W × K solutions are known, the dependence of these solutions on a particular choice of initial conditions is a fundamental problem (for example the question arises: is it possible to obtain a FRW × K model from arbitrary homogeneous anisotropic initial conditions?). Maeda [6 ] in his numerous works developed the idea that the solution F R W × K is an attractor, i.e., that it is represented by an attracting critical point of the corresponding dynamical system, for which the dimension of the invariant attractive manifold Watr is equal to the dimension of the phase space. This idea is connected with the intuition that a ..sufficiently old Universe evolves independently of its initial conditions imposed onto it in the far past (i.e., the multidimensional universe has no hair [ 7 ] ). Maeda assumed that both macro- and micro-space were some particular spaces, i.e., both depended only on time, the physical space being maximally symmetric and the internal space - the Calabi-Yau manifold (for details see the appendix). The effective lagrangian leading to compactification of the internal space is described by its radius b and by a set of real scalar fields ¢~, i= 1..... N (the socalled dilatons). The higher dimensional Einstein

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equations are reduced to four-dimensional ones with an additional scalar field ON+ 1~ In b. By using the method of dynamical systems, we recover (see the appendix) Maeda's theorem which states that the existence of a local minimum of the potential function is a sufficient condition for the stability of the FRW × K solution. Let us note, however, that the method used by Maeda is nothing but an investigation of the asymptotic stability in a strictly determined class of solutions, namely those defined by the choice of the metric [8 ]. In the present work we investigate the problem of sensitiveness of solutions admitting stable FRW × K configurations with respect to the choice of the initial geometry of the model as its initial condition in superspace. We do not investigate the stability of particular solutions of the respective dynamical system in the phase-space but we discuss the stability in superspace where the solution is represented by a geodesic. First, we can reduce the problem of studying the hamiltonian system describing the dynamics of multidimensional cosmological models to the problem of investigating the geodesic flow on some pseudoriemanian manifold. Afterwards, we are able to investigate the behaviour of nearby geodesics in this space. Such an approach has been developed by Savvidy in his works concerning the chaotic behaviour in YangMills theory, and by Gurzadyan and Kocharyan [9 ]. The method used by us is based on investigating the behaviour of nearby geodesics on different metric manifolds which are configurational spaces of a hamiltonian system [10]. This method is global in the sense that it deals with the behaviour of nearby geodesics, and not with some points on them. Two points on different geodesics can be close to each other whereas the geodesics themselves can diverge [ 11 ].

2. Multidimensional model as a four-dimensional model with a scalar field

Let us assume that the metric of our multidimensional model has the form

dg=b-~g ..... dx m dx"+b2~,~u(y) dy M dy N ,

(1)

where D is the dimension of the internal space, the m e t r i c g?.lN of the internal space does not depend on 712

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the coordinates of the physical space having metric gin,. The factor b-/~ of the metric of the physical space has been introduced to obtain the correct Einstein action in four dimensions [ 6 ]. Ansatz ( 1 ) for the metric enables us to reduce the ( 1 + 3 +D)-dimensional lagrangian of the system to the ( 1 + 3)-dimensional one:

I=f~dl+3+Dx-_f~d'+3x, where ~ and ~ are ( 1 + 3 + D)-dimensional and ( 1 + 3 )-dimensional lagrangians, respectively and, as in the case discussed by Maeda,

where 0i, i= 1, ..., N a r e real scalar fields, b is the scale factor of the internal space, 17is the effective potential of the ( 1 + 3 +D)-dimensional theory. The effective lagrangian in ( 1 + 3 ) dimensions is Lf= ½ x f ~ R + 5°~+' , Z

i:1

( v 0 i ) 2 + o ( 0 , ..... 0=+,)

,

with ON+ ~~ In b, R is the curvature scalar computed with respect to the metric gin, (x), - g = det gin,. After the dimensional reduction of the ( 1 + 3 + D ) dimensional system with the potential 17, we obtain the system describing four-dimensional gravity and N + 1 scalar fields with the potential U.

3. Minisuperspace of multidimensional cosmological models

Without loss of generality we shall assume that in the ( 1 + 3 +D)-dimensional space-time with metric ( 1 ), all real scalar fields 0~, i = 1.... , N are negligible. Our multidimensional system will then be equivalent to four-dimensional gravity with one scalar field O N + 1 ----- : O ~ In b, i.e., at the level of action: I = Igrav q- 10. We shall also limit our considerations to the case when the spatial part of the metric gmn is the metric of a compact maximally symmetric three-manifold S and it depends on a single scale factor a(t): h0= a2a2(t)~i. This choice is pragmatic and it is a con-

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sequence of the fact that in minisuperspace models the infinite number of gravitational and material degrees of freedom is reduced to a finite number. As a convention, we choose the normalization constant so, that t72=(3mp/4~z) f ~ l / 2 d 3 x , where mp is the Planck mass. For simplicity of equations, we shall use the scalar field Z = a : ¢ and the potential U ( Z ) = (4rw2/3mp) O(~) instead of O and t.7. Four-metrics close to the model in question are the following:

pseudoriemanian manifold. The action for system (5) is

ds2= - N 2 dt2 +ha dx i dx j .

u'~= d x U ds ,

(2)

(3)

where a=ln a,

~D~ =½e-~"(-P~.+P~)

v ~= dxa/dz=pag ab ,

~ b c _ _ .~c ~Jabt5 --Va ,

,

where s is a certain parameter. Let us also denote by W ÷ and W - the following regions in the minisuperspace: W + = {x: V(x) > 0}, W - = {x: V(x) < 0}. since of= ½gabvad'+ V(X)=0, thus in the region W - one has: gabVaVb= --2 V(X) > 0. For the extremal curve ? of action (6) one obtains

8I=O=S f padxa=S f ga,,vavh dz

- e 3 " [ U ( Z ) - ½k e -2" ] .

Y

The variation with respect to a, ~, P,, Po gives us dynamical equations and the variation with respect to N a constraint condition ~ADM= 0. k is the curvature of three-dimensional spatial sections: k = + 1, if S is a three-sphere or a three-sphere factorized by a certain discrete group S3/F, k= 0 if S is a three-torus or another compact form of a fiat space, k = - 1 ifS is a factorized pseudosphere: p S 3 / F (for a discussion of compact forms of maximally symmetric spaces see ref. [ 13 ] ). The minisuperspace M is a two-dimensional space with coordinates a and Z. When N is constant on hypersurfaces of homogeneity the metric of the minisuperspace F(N) is dQ2 = 0"N -J ( - a d a 2 + a 3 dz 2) ,

(4)

and it is flat for N independent ofx.

4. The reduction of the hamiltonian system to a geodesic flow Now, we shall reduce the hamiltonian system with the hamiltonian

of = ½gabp,p,, + V(x) ,

(6)

Acti~ a (3) can be reduced to (6) if one imposes the condition d z = N dt. Let us introduce the following notation:

Pa =gab vb

The A D M action o f our system is [ 12]

I= f P, da + P, d(~-- N~ADM dt ,

I = f pa d x a - ofd'c •

( 5)

and with the constraint of = 0 to a geodesic flow on a

Y

= 5 f x / - 2 V(x) x/gabv"vb d r Y

=xf2 8 f ~ - V(x)gah dx a dx b 7

=v~fds.

(7)

Y

Hence, the least-action principle (7) is equivalent to the problem of geodesics on a manifold with the metric Gab = -- V(x)gab. One also sees that the parameter s is defined so as d s = x / ~ ( - V ) d z and I[u[l= Gab uau b= 1. An analogous consideration concerning the region W + shows that the problem of the behaviour of system (5), in general, can be reduced to the problem of geodesics on a manifold with the metric Gab = [ V(X)Igab, and d s = v / ~ [ V(x) I dr, Ilull = - s g n V(x). It can easily be seen that the representation obtained o f the hamiltonian system as a geodesic flow is a simple consequence of the Maupertuis principle. This enables us to represent trajectories of the system

dqa/dt= OH/Opa, d p a / d t = - OH/Oqa 713

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in the phase space {q~, Pa; a = 1.... , n} as geodesics on a riemanian space with the metric

ds2=[E-U(qt,...,q")] ~, (dq") 2 a

To investigate the stability of the geodesic flow, i.e., the global stability of solutions, we use the Jacobi equation which, in the Fermi basis {Ei}, is

d2zi/ds2 + K~zJ=O ,

(8)

where z=z'Ei is the geodesic deviation vector, Kj the two-dimensional curvature: K)= (E i, R(u, Ej)u), {E i} is the basis dual to {Ej}. The Fermi basis is chosen so that one basis vector is tangent to a geodesic and the other are orthogonal to it. After the reparametrization s ~ z deviation eq. (8) assumes the form

d2z'/d'g2"}-~('r) dz'/dr + z~og}= 0 ,

(9)

where 7(r)=-dlnlVI/dz,

og~=2V2Kj.

(10)

29 D e c e m b e r 1988

5. Stable compactification in minisuperspace As is shown in the appendix, the asymptotically stable configurations correspond to H=Ho, ~=0o, U ' ( O ) I ~ = 0 . From eq. (A1) one can see that this critical point lies on the boundary of the condition k/a 2= O, i.e, without loss of generality we can assume that k = 0 and consequently use formulae ( 13 ). The Jacobi equation (9) now assumes the form

~=3H2+(U'/U)xk-(U"-U'2/U)z,

where the dot denotes differentiation with respect to r. We shall investigate the behaviour of geodesics when

H~Ho=x/Uo/3,

x--,Xo = 0 .

Eq. (14) can be written as

2-3Ho~_+ U" (Z)z=O.

KI =(E~,R(u,E~)u) =(EI, K[llull2El-(U,E,)u])=gllu[I 2,

(11)

and the problem is reduced to the study of geodesics on the space with the metric G~,= Grl~,,where

G=e6"lU-½ke-2'~l,

r/~b=diag I I - 1, 111 •

(12)

Formulae (10), ( 11 ) and (12) are simplified if k = 0 , and one obtains [ V[ =e3" I U[ ,

G=e6"[ UI ,

7(z) = - 3 H - d In ] U[ ~dr, 1 [31nG G -

K(z)-- 2

1 ~lnlUI 2 e6'~lU] '

og(z) = UI~ In IUI =U"-U'2/U,

(13)

where H is the Hubble function, E ] = - 0 2 / 0 ~ x 2 + a2/OZ 2, U' =dU/dz. 714

U" ( , ~ ) =

°" 2

d~ 2

,

d2z 1 d2z d,L.2 -- N2 dt 2 .

Eq. ( 15 ) now is ~+a~-flz=0.

I gl = e 3c~ ] U - ½ke-2"l ,

(15)

In the process of compactification of the microspace to a constant size: b~bo=const., N-~No=bffD/2= const, and 1 dU(~b)

In the above considered case, when the hamiltonian is given by formula (4), the dimension of the configurational space is equal to 2, and one has

(14)

(16)

where o~= 3HoNo, fl= - N~/_3" ( 0o ) / a 2. Now the dot denotes differentiation with respect to t. Eq. (16) may be reduced to the form of the dynamical system

~=y ?~=flz+~y.

(17)

Eq. ( 16 ) has the form of an oscillator equation with the damping force c~ and the exciting force flz. Let us notice that: (1) A positive value of Ho plays the role of an "antifriction" i.e., it destabilized the system. (2) The exciting force depends only on position z, and is strictly linear. (3) If a = 0 then, for fl=0, we have a mathematical pendulum in the lower equilibrium position, and for fl> 0 a mathematical pendulum in the upper equilibrium position. Every point of the plane (fl, a ) (see fig. 1 ) represents the evolution of the geodesic deviation. The type of critical points is determined by the eigenvalues of the linearization matrix. If fl is not constant in time,

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_______~\ u~,tabto ~ c<

Ons~o.bte~

.saddtes

v

saddLes

kn~l's Fig. 1. the structure of the phase space may change. The only distinguishing case is when a = 0 and fl< 0, because only then may nearby geodesics have the limited deviation. In order to see this, it is sufficient to choose the Lapunov function in the form

v(z,y)=½yZ+ f f(t) dt. 0

This function is positively defined and dr~dr= 0. the point z=y= 0 is stable by the Lapunov theorem. In the general case, in the neighbourhood of a local minimum of the potential, the dynamical system may be approximated by its linear part, on the strength of the theorem of the iinearization of a nondegenerate system. By substituting these formulae into the equation of geodesic deviation, one can show that for t-,oo, periodic trajectories occur.

6. Conclusions It was proposed [ 12 ] that inflation with exponential evolution describes the transition from a higher dimensional cosmology to the effective four-dimensional cosmology (for alternative ideas see refs. [ 13,2 ] ). In this context inflation could be regarded as a purely gravitational mechanism. According to what we have proved, if such an inflation state had been reached asymptotically (H--,Ho)

29 December 1988

then the corresponding solutions would have been globally unstable. This makes us hesitant to regard inflation as a dynamical effect of extra dimensions. There are many examples known in the literature ofcompactification to the Minkowski X K (static internal space) - space. The following cases belong to them: (a) The classical vacuum F R W ( k = 0 , - 1 ) X T D models and their generalizations are compactified to the anisotropic case Bianchi V X T D, Bianchi I X T D [3,14] (see also the appendix). (b) FRW × Calabi-Yau space models with a positive Casimir energy at low temperature approximation [ 15 ]. The compactification mechanisms of type (b) are globally stable, whereas the mechanisms of type (a) are unstable. When the effective action has a local minimum, as it is in the case of six-dimensional Einstein-Maxwell theory or in models with monopole and low-temperature quantum effects [ 5 ], such solutions are globally unstable. The geodesics expand exponentially in minisuperspace which is an evidence of the existence of chaotic behaviour in minisuperspace [9 ]. The problem of the chaotic behaviour in minisuperspace can have consequences for a quantum chaos since in Hawking's approach the boundary conditions for the Wheeler-de Witt equation are given in minisuperspace. Because solutions should depend on the initial conditions in a continuous way, chaos in minisuperspace would have some consequences for quantum cosmology [ 16 ].

Acknowledgement The authors are grateful to Professor M. Heller and M. Biesiada for reading the manuscript and for stimulating discussions. This work was partly supported by the Polish Interdisciplinary Project CPBP 01.03.

Appendix In this appendix the theorems of Maeda [6] will be recovered by using the method of dynamical systems. We shall consider the case of constant potential, which corresponds to vacuum cosmological 715

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models, and it will be shown that the single attracting critical point can never exist in the phase space. The basic equations are the following: Einstein's equations:

--2Ho 0 "

L= H2+k/a 2

=

p + ~ Z ¢~+U(Ol,...,ON+,)

,

(A1)

i=l

f

)

~v+l

2([-I-k/a2) = - t p+p+ ~1 0~

,

(A2)

29 December 1988 0

...

0

...

0-

0

0 0 : 0

-h

A

where h = (0 ~ U) is the hessian, and A = - 3 H o ~ . Let us consider the eigenproblem of the matrix L: d e t ( L - 2 ~ ) = - (2Ho + 2 ) ( - 2 ) u+~ d e t ( A - h / 2 )

the equations o f motion o f the scalar fields

b~+3Hb,=-OU/O¢i,

i = 1 ..... N + I ,

=-2(//o+2) (A3)

×det[ -h-

( - 1 ) N+I (3Ho + 2 ) 2 ~ ] = 0 .

where the metric o f the (4 + D)-dimensional spacetime is of the form

By denoting z = 2 ( 3 H o + 2 ) we obtain

ds2=b-O(t){-dt 2

Hence, eithe," ). = - 2//o or det ( - h - z~ ) = 0 is satisfied for eigenvalues zl .... , ZN+ ~. If the hessian is positively defined the potential has a minimum, all zi, i = 1.... , N + 1 are positive. One has the following relations for every z,:

+a2(t) [dr2/(1-kr2)+r

2 d~22])

+ b 2 ( t ) gA,N(Y) dY M dY u where gaiN is the metric of the internal space and it does not depend on coordinates of the physical space, D is the dimension of the internal space, the field q~x+~ ~ In b, where b is the scale factor of the internal space, H=a/a the Hubble function of the physical space with a being its radius, p and p are pressure and density in the macrospace. Real scalar fields ~ ..... CN are given in the physical space with Ubeing their potential. One can construct an autonomous dynamical system corresponding to eqs. ( A 1 ) - ( A 3 ) . For simplicity, but without loss of generality, we assume that p=p= O. The system in question assumes the form 1 N+I

/:/= -s-v- ]. ,__E,x~ + ~ v(¢,,, ..., ¢,N+,).

~=-3Hx,-OU/O~,, System

(A4)

has

i = 1 ..... N + I . the

critical

(A4) point

H~=

I U ( ¢ I io, ,,,, CN+ i[o) '

¢,1o, 0U (¢~1o)/0¢, = 0 . The linearization matrix is a ( 2 N + 3 ) × ( 2 N + 3) matrix and has the form 716

d e t ( L - 2 ~ ) = ( - 1 )N+2 ( 2 H o - t - 2 ) d e t ( - h - z ~ )

=0.

(3Ho + 2 ) 2 + ~ o , = 0 and 2, = ½ ( - 3Ho + x / 9 H 2 - 4o9i ) . One can see that if o9~> 0 then all ;t~ are negative which means that the dimension of the attractive manifold W a~r is equal to that o f the phase space.

References [ 1 ] E. Witten, Nucl. Phys. B 186 ( 1981 ) 412; A. Salam and C.N. Pope, in: Supersymmetry and supergravity '82, eds. S. Ferrara, J. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983 ). [2] D. Shader, Phys. Lett. B 155 (1984) 1378; Phys. Rev. D 30 (1984) 756; R.B. Abbott et al., Phys. Rev. D 30 (1984) 720; D 31 ( 1985 ) 673; E.W. Kolb et al., Phys. Rev. D 30 (1984) 1205. [3] M. Demianski et al., Class. Quant. Gravit. 3 (1986) 1199. [4] M. Szydtowski, J. Szcz~sny and M. Biesiada, Class. Quant. Grav. 4 (1987) 6; P. Candelas and S. Weinberg, Nucl. Phys. B 237 (1984) 397; M. Yoshimura, Phys. Rev. D 30 (1984) 344; Prog. Theor. Phys. 73 (1985) 849,942;

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Y.S. Myung and Y.D. Kim, Phys. Lett. B 145 (1984) 45; J.A. Stein-Schabes and M. Gleiser, Fermilab preprint 85/

157-A (1985); Y. Okada, Phys. Lett. B 150 (1985) 103. [5] F.S. Accetta, M. Gleiser, R. Holman and E.W. Kolb, Nucl. Phys. B 276 (1986) 501. [6] K. Maeda and H. Nishino, Phys. Lett. B 154 (1985) 358; B 158 (1985) 381; K. Maeda, CLass. quant. Grav. 3 (1986) 233; K. Maeda, in: Proc. Intern. Symp. on Particles and the Universe (1985); in: Proc. Fourth Marcel Grossmann Meeting ( 1985); K. Maeda, Phys. Lett. B 166 (1986) 59; SISA Trieste, preprint 63/85/A ( 1985); K. Maeda and P.Y. Pang, Phys. Lett. B 180 (1986) 29. [7] J.D. Barrow, Phys. Lett. B 187 (1987) 12.

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[8] J.D. Barrow and H. Sonoda, Phys. Rep. 139 (1987) 1. [9] V.G. Gurzadyan and A.A. Kocharyan, Zh. Eksp. Teor. Fiz. 93 ( 1987 ) 1153; preprint EFN-921 ( 72 )-86 ( 1986 ). [ 10] V.F. Arnord, Mathematical methods of classical mechanics (Nauka, Moscow, 1979) [in Russian]. [ 11 ] S.W. Hawking and F.R.G. Ellis, the Large scale structure of space-time (Cambridge U.P., Cambridge, 1973 ). [ 12] Q. Shaft and C. Wetterich, Phys. Lett. B 129 (1983) 387. [13] M.B. Gavela, Phys. Rev. Lett. 51 (1983) 931. [ 14] M. Demiafiski et al., submitted to Class. Quant. Grav. [ 15 ] K. Maeda, preprint Trieste IC/86/316. [16]J.B. Hartle and S.W. Hawking, Phys. Rev. D 28 (1983) 2960; S.W. Hawking, Nucl. Phys. B 239 (1984) 257.

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