Volume 167B, number 2 PHYSICS LETTERS 6 February 1986

HIGHER-DIMENSIONAL BIANCHI COSMOLOGIES

D. LORENZ-PETZOLD

Fakultiit ftir Physik, Universitiit Konstanz, D - 7750 Constance, Fed. Rep. Germany

Received 18 November 1985

We present new higher-dimensional Bianchi cosmologies of class A. Our solutions given are of type M” = RX M3 XTD,

where N = M3 are of Bianchi types I, II, VI,, VIIO, VIII and IX. This spectrum of solutions includes the higher-dimensional

versions of the Kasner solution and the mixmaster universe with stiff matter content.

1. Introduction. In the last few years, there has been a substantial and increasing interest in the study of cosmologies with more than four dimensions (see

ref. [ 11 for references). These higher-dimensional scenarios are based on various Kaluza-Klein theo-

ries. The most promising candidates today are the su- perstring theories in d = 10 dimensions [2,3] . The cosmological and astrophysical implications of these theories have been recently discussed by various au- thors 14-I 2 ] . According to the Kaluza-Klein pic- ture of the universe, we are interested in curved space- time manifolds of type Mn = R X M3 X MD, where M3 corresponds to the various anisotropic Bianchi types I-IX (including the isotropic Friedmann- Robertson-Walker (FRW) models as special cases) and MD denotes some compact space (see, however refs. 113,141).

By setting D = 6 we are faced with the problem of classifying six-dimensional spaces. The construc- tion of the so-called Calabi-Yau manifolds is a very exotic topic of current research [IS-181 . Some ap- plications to the cosmological problem have been given by Maeda [ 11,121. Some ten-dimensional cos- mologies with toroidal M6 = T6-space are discussed by Freund and Oh [9]. The solutions given are iso-

tropic in (1 + 3)dimensions with some anisotropic behaviour in the extra dimensions. In this paper we are interested in higherdimensional cosmologies which are anisotropic in both the physical as well as the extra space. Such cosmologies have been recently discussed by Lorenz-Petzold [19,20] and Demaret

0370-2693/86/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland Phvsics Publiahine Division)

et al. [21] on the basis of N = 1, eleven-dimen- sional supergravity. Some isotropic solutions in ten-dimensional supergravity are also known [22-241 .

It can be shown that in the Scherk limit o’ + 0 (a’ is the inverse string tension) we rediscover higher- dimensional Einsten gravity coupled with a scalar field @ [9,11,12]. However, such a scalar field is known to be equivalent to the incorporation of “stiff’ matter in the pure Einstein theory. Since no cosmolog- ical solution is known for the general case (Y’ f 0, we consider our exact solutions (see below) as some ap- proximations valid at some stages in the evolution of the multi-dimensional string universe. By setting N = M3, where N = I, II, VI,, , VII,, VIII and IX are the various Bianchi types, MD = To = SL X . . . X S, we assume a metric of the form

ds2=q//‘u”, /J,v=O ,..., D, Q~~=(-l,l,..., I),

(la)

where

c”=oo=dt ’ , o’=Riwi, 0” =rn?h” (no sum),

(lb) and Ri = R,(t), r, = r,(t) are-the cosmic scale func- tions, i = 1,2,3; m = 4, . . . . D. The differential one- forms oi are defined as in table 8.2 of ref. [2.5 ] for the various Bianchi types. In addition, we introduce the variable rD by rD = IIr, . In the next section we present our corresponding field equations and the gen- eral solutions.

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Volume 167B, number 2 PHYSICS LETTERS 6 February 1986

2. Field equations and solutions. The field equa- tions to be solved are given by

(In RF)” = r20 [(nj R; - ~QR:)~ - n;R;]

+ 2g2@ - p)/(n - 2), (2a)

(In r,)” = g2(e - p)/(n - 2) (2b)

(lng)12 -c Hi” - C/z;

= (r92){n2R4 t n2R4 + u2R4 11 22 33

- 2[qn2(RlR2)2 +nln3(RlR3)2

+ n2n3(R2R3)21 1-t 2g2,, (2c)

dt(etp)(lng)‘=O,

p=(y- l)e, 1<7<2, (2d)

where Hi = (ln Rj)‘, h, = (In I~)’ are the Hubble parameters,g=R%D,??s =R,R,R,,dt =gdq, ( )’ = d( )/dn, (i,i, k) are in cyclic order and the nj are the structure constants of the various Bianchi types, defined in table 1.

If we define the new variables Si by Sj = Ri ,D’2) then we obtain from (2a) and (2b) the threedimen- sional system

(ln Sf )” = (nj Sf - nkSi)2 - nf R: . (3)

However, this is not possible for eq. (2c), which gives additional information not present in the three-dimen-

sional case (h, = 0). The “stiff’ matter is character- ized by y = 2, i.e. p = e. From the conservation law (2d) it follows that e = mgv2, m = const.

Let us see what happens in the most simplest case nj = 0, i.e. for the Bianchi type-1 model. From eos.

Table 1 __-__--___

n1 n2 n3 Bianchi type -

0 0 0 type I 1 0 0 type II 1 -1 0 type VI, 1 1 0 type VII, 1 1 -1 type VIII 1 1 1 type IX

_____

158

(2a) and (2b) we obtain the solutions

R, = exp (Ki$, rm = exp (k,v), Kj, k,,, = const.

(4a)

K+k=l, K=CK,, k=Ck,, (4b)

and from (2~) the additional constraint

c(Kf tki)= 1-2m.

By setting

(4c)

qi=Ki/(k+K), Pm =k,I(k+K), (sa)

we obtain the generalized pseudo-Kasner solution

Rj=t% 3 r =tPm m WI

C(qjtpm)=C(qf tpi)= 1-2m. (5c)

The vacuum case m = 0 has been first considered by Belinskii and Khalatnikov [26] and later by Chodos and Detweiler [27] . The special case q+ = q1 = q2 =q3,p_ =p4 = . . . = pD yield the relations

q+=$[l -(D- 3)p_], D>4, (6a)

P_ = (l/D){1 + [3(D - l)/(D - 31 lj2}, (6b)

from which we conclude that we may have an iso-

tropic solution in (1 t 3) dimensions, while the “radi- us” of the extra space MD contracts (from a certain time t = to) at an increasing rate and would thus not be observable at later times. In (1 t 4) dimensions we may have q+ = -p_ = 3 and the corresponding solution (Sb) behaves in (1 t 3) dimensions as the radiation-dominated FRW model with R = t2i3r, 7 = 413.

As noted in the introduction, the incorporation of “stiff’ matter (7 = 2) is entirely equivalent to the pres.

ence of a scalar field @. Consider the Kasner solution in (1 t 4) dimensions with

Rj=tsj, c,=c,z=I, j=l,2,3,4. (7a)

Introducing the transformations

si = (61’2pj - q)/(6l’;! - q), s4 = 2q/(61’2 - q),

qf6 , ‘I2 B=qlnt, q=const., i=1,2,3, (7b)

Volume 167B, number 2 PHYSICS LETTERS 6 February 1986

it follows that

R =tPi, &$=I, c p;=1-q2, i PC)

which is of the desired form (5~). The Kasner solu-

tion (m = 0) is known to be completely anisotropic in (1 t 3) dimensions. In contrast, the incorporation

of stiff matter gives an isotropic FRW model with q+ = m = 4 for D = 0. (It has been stated by Shikin [28] , that the p = e solution does not become isotropic; this is not true. For comparison see the special L.R.S

solution(l2.ll)ofref. [25] withA=l,m=h(2th)/

(1 +2x)?) The main interest in our solution (5b), (5~) comes

from the higher-dimensional version of the so-called mixmaster universe [29], which is of Bianchi type IX. In (1 + 3) dimensions, the Kasner solution arises ex- actly when the right-hand sides of eqs. (2a) and (2~) are zero. The effect of the curvature terms on the right-hand sides is to permute the evolution of the Ri = R,(t) through a series of solutions of type (5) but with different Kasner exponents. This leads to the well-known chaotic behaviour of the solution near the initial singularity (see refs. ]30,3 11 for latest developments in (1 t 3) dimensions). However, the situation may be different due to the influence of the extra dimensions. This can be seen by performing the transformations

ui=Pit;cPm, Ri = t”i, ‘0 = (4iY Pm),

a= l,...,D, (8a)

from which it follows that

Cl$= 1 t+.zPm,

,&;=I -(CP;)t$(CPm)2+~P,,,, @b)

3 CP, 2.0. u1u2+u1u3+u2u3 =4 ( 1

(8~)

In (1 t 3) dimensions we have P,,, = 0 and thus one of the Kasner exponents Ui = qi must be negative. This leads to the chaotic behaviour in the original mix- master model. In the higher-dimensional case we have Pm # 0 and thus it is possible for all the Ui to be posi- tive, from which we conclude that the interaction with some higher-dimensional space may prevent the

chaotic behaviour near the initial singularity. Our re-

sult is in accord with the recent studies 132-361 of higherdimensional mixmaster cosmologies.

In the next step we are interested in explicit solu- tions for the more general models of types N = M3, where N = II, VI,, VII,, VIII and IX (Class A in the Bianchi classification; for higher-dimensional class B

solutions, see e.g. refs. [37-401). (i) N = II:

(ln(rDR1R2))” = 0, rDRlR2 = ew(k12d, (W

(ln(rDR,R3))” = 0, rDR1R3 = ew(k13v), (9b)

(In .z)‘~ - B2 t z2 = 0, z = B sech(Bv), (9c)

rm = exp(k,r& rD = exp (kv), (W

where

kii 3 k m = const., k= Ck,,,

B2 = 4k,,k,, - k2 - 2 Ck; - 4m,

z =yrD, y =Rf.

We obtain

Rf = B exp(-kn)sech(B7]),

Ri = B-’ exp[(2k12 - k)q] cosh(Bv),

Ri = B-l exp[(2k13 - k)q] cosh(B&

(W

‘rn = exp (k,rl).

(ii) N = VI&$r = 0):

(In rDR)” = 0, rDR = exp (bv),

(ln rDS2)” - 2(rDR2)2 = 0,

(90

(lOa)

(lob)

(ln S)’ =A-1(rDR2)2 - B/2A,

where

(1Oc)

R =R1 =R,, S=R,, A =2b -k,

B=2b2-k2-Cki--2m, b=const.

nPp denotes the anti-symmetric part in the Ellis-

MacCallum [41] decomposition of the structure con-

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Volume 167B, number 2 PHYSICS LETTERS 6 February 1986

stank The solutions can now be easily completed in terms of the Ellis-MacCallum parameter u = R2 :

R2 = u, rm = Uk,/2(b-k), ,.D = uW(b-k),

k=C k,, (IO@

s2 = u -B/2A(b-B)

X exp { [ l/A(2b - k)] u(~~-~)‘(~-~)}, b f k.

(1Oe)

(iii) N = VI,, VII0 (rifle # 0):

(ul + u2 + u)” = 0, u1 +ul tu=b(71-n0), (lla)

vn? ” =O, vm=k,q, u=&,=k=~k,,(llb)

u” t 4 exp [2b(77 - ~$1 sinh (u) = 0,

2.4 = 2(u1 - u2),

bu; = -u;u; -B

(llc)

+ exp(2kn)[exp(2u1) - 6 exp(2u2)] 2/4, (lld)

where

Rj = exp(u$ rm = exp(u,),

B=k(b-k)-$k;-m, 6 =n2.

After solving eq. (11 c) for u = u(n), the solutions would be completed by

u1 =(A -u)/2tu/4, u2=(A -@2-u/4, (12)

where A = b(q - qo), and the solution u3 = u3(r)) is

given by eq. (1 Id). By setting k = k, = 0, we redis- cover the (1 t 3)dimensional system first discussed by Lorenz-Petzold [42] in a complete manner. (It has been stated by Barrow and Stein-Schabes 1431 that there is no Bianchi type-VII, model in vacuum or con- taining fluid; our model with 6 = 1 disproves this statement.) Introducing the time variable 4 by

E = (2lb)exp [WI- vo)l , (13)

we can transform the system (1 l), (12) to obtain

u1 = ln[(Cb/2)(b-k)‘2b] -kg,, t u/4, (14a)

160

u2 = ln[(@/2)(b-k)‘2b] - kqo -u/4,

%n = ln[(gb/2)k”‘b] t k,qO,

Ei t fi/t t sinh(u) = 0,

ti3 = -[(b - k)2 t4B]/4b2g

(14b)

(14c)

(14d)

+ t {C2 t 2[cosh(u) - 6]}/16,

where ( )’ =d( )/d.!j.

(14.e)

Our equation (14d), which is crucial for the whole

system (14) was derived for the first time by Belinskii and Khalatnikov [44] in the context with the Bianchi type-IX model in (1 t 3) dimensions (for the Bianchi

type-VIII model, see refs. [45,46]). It was shown that near the initial singularity the function u3 may be neglected compared with the two others u1u2 in the right-hand sides of eqs. (2). Thus their approximative field equations for types VIII, IX are the same as our exact field equations (in (1 + 3) dimensions) for types VI,, VII,. This has been first observed by Lorenz- Petzold [42] and independentlv by Jantzen [47]. A more recent study of eq. (14d) was given by Khalat- nikov and Petrovski 1481 for the Bianchi types VIII, IX in terms of elliptic functions. Here we present an- other solution in terms of a particular form of the Painleve transcendent of type III [49].

If we put

w = exp(u), z = g2/4, w = w(z), ( )’ = d( )/dz,

eq. (14d) becomes (lsa)

w” = w’2/w - [w’ + (w2 - 1)/2] /z, (15b)

which defines the third Painleve transcendent w = w(z), It is now an easy matter of calculation to reexpress the remaining functions Ui in terms of w.

(iv) N = VIII, IX:

(lng)’ t 8g2 = 0, (16a)

g=sech(q-ql), 6= 1,

g = csch(q - ql), 6 = -1, (16b)

‘M =exp(k,q), rD =exp(kq), k= Ck,, (16~)

(ln z)‘~ - B2 t z2 = 0, z = Bsech(B(q - qo)), (16d)

Volume 167B, number 2 PHYSICS LETTERS 6 February 1986

where PI

g=rDRS, R =R, =R,, S=R,,

B2=4-k2-2Ck;-4m,

]31

[41 [51

qi = const., z = rDs2, 6 = n3.

The solutions are completed by

S2 = B exp(-k$sech(B(q - Q,,)), rm = exp(k,n),

(16e)

[61 [71 181

PI [lOI

R2 = B-l exp(-kn)sech2(q - nl)cosh(B(q - q,)),

type IX, (I6f)

= B-1exp(-kq)csch2(n - rll)cosh(B(n - no)),

1111 1121 iI31 [I41 P51

type VIII. (16g)

3. Conclusions. We have given a complete deriva- tion of higher-dimensional cosmologies of Bianchi

class A. Only the Bianchi type I vacuum solution (5) and the Bianchi type IX solution (16e), (16f) was known (see refs. [34,35] ; we do not believe that the D = 6 = 1 solution of ref. [35] is entirely correct (see the restriction given by 82)). The k = km = 0 solu- tions (9) and (16e), (169 reduce to the (1 + 3)dimen- sional solutions first given by Taub [50], However, the higher-dimensional solutions given in this paper allow for a very different behaviour. For instance, the Bianchi type II solution (9) allows for the condi- tions k = 2k12 or k = k13. It follows that one of the

factors exp [(2kii - k)v] vanishes in the expressions

for the cosmic scale functions R2 or R3 . This would have a serious influence on the Hubble parameters in physical space. (Note that the solutions (6b) of ref. [34] do not exist.) The same remarks apply also to the more general Bianchi types VI,, VII0 , VIII and IX. To proceed in the topic of higher-dimensional su- perstring cosmologies, it would be nice if in the near future the explicit form of the Calabi-Yau spaces

could be found.

P61

1171

1181

iI91 PI wj

WI ~231

1241

1251

[26] V.A. Behnkskii and I.M. Khalatnikov, Zh. Eksp. Teor. Fiz. 63 (1972) 1121 [Sov. Phys. JETP 36 (1973) 5911.

[27] A. Chodos and S. Detweiler, Phys. Rev. D21 (1980) 2167.

[28] I.S. Shikin, Dokl. Akad. Nauk SSSR 179 (1968) 817 [Sov. Phys. Dokl. 13 (1968) 3201.

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