Physics Letters B 277 (1992) 393-397 PHYSICS LETTERS B North-Holland

Supersymmetric general Bianchi type IX cosmology with a cosmological term

Robert Graham

Fachbereich Physik, Universitiit Gesamthochschule Essen, W-4300 Essen I, FRG

Received 4 November 1991; revised manuscript received 13 December 1991

The covariant action of the supersymmetrically extended spatially homogeneous general Bianchi type IX cosmology is given including time reparametrization invariance, inducing the hamiltonian constraint, and the constraints on the supercharges. A supersymmetric cosmological term is also included. A quantization respecting the supersymmet~' and the conformal invariancc of the superspacc of geometrodynamics is given. The problem of tbe initial condition is naturally solved in this description by demanding that the wave function does not diverge if the anisotropy variables are taken to infinity for fixed scale parameter.

Among the spatially homogeneous cosmological so- lutions of general relativity those of Bianehi type IX (i.e., those with SO (3) as a simply transitive isometry group of the three-geometr3, [ 1 ] ) are of particular in- terest because they are believed to exhibit the full complexity of the cosmological singularity {2-4]. As a consequence time-dependent Bianchi type IX met- rics are among the best studied cosmological mod- els, both classically [1-5] , and quantum mechani- cally [6-10] . Recently it was shown [10] that the diagonal Bianchi type IX model without cosmologi- cal term has a natural supersymmetric extension, and as a consequence, an exact analytic solution of the Wheeler-DeWitt equation could be given. Supersym- metric extensions have been given before for the spa- tially fiat Bianchi type I model [ 11 ] and the isotropic Friedmann model [ 12 ] which is also of Bianchi type IX.

In the present paper we (i) give the supersymmetric extension for the general non-diagonal Bianchi type IX model, (ii) exhibit the associated first order action in- cluding all constraints, (iii) extend the model further by including a cosmological term within the supersym- metric framework, and (iv) discuss the quantization of this model respecting the conformal invariance of the superspace of geometrodynamics.

We begin with the action of general relativity. Writ- ing the space-time metric as

ds 2 = - ( N 2 - h, S N i ) d t 2 + 2~ \~ dx i dt

+ gu dx~ dxJ, (1)

thc first order form of the action in units with G = c = = 1, i s [ 4 ]

' / ( S = T - ~ . d4x zcij OgiJot - NtI(zeij'giJ)

-.A,]H i (#Y, gij ) ) ,

with

I I = ((3~ g)' /2(Gijk, nOT~k'-(3'R + 2 A ) ,

[t i = --21zik;k ,

(2)

(3)

1 Gijkl = ~ (gikg# + g~tgjk -- gijgkt). (4)

Here (3)g, (3) R and the covariant derivative refer to the thrce-geometI3' described by gij. ,4 is the cosmo- logical constant. In (2) the Lagrange multipliers N, ~ , and the canonical variables ga, 7tii are to be varied.

Let us now restrict this variation by assuming the metric to be spatially homogeneous and of general Bianchi type IX form, with a scale factor a(t) = (6ze) -t/2 expa ( t ) :

0370-2693/92/$ 05.00 Q 1992-Elsevier Science Publishers B.V. All rights reserved 393

Volume 277, number 4 PHYSICS LE'I'I'ERS B 12 March 1992

1 ds 2 = - N Z ( t ) d t 2 + ~ e x p [ 2 a ( t ) ] R k j ( t )

× (exp [2/3 (t) ] )k/RIj ( t ) a i a j . (5)

Here the a ~ are basis one-forms on a three-sphere satisfying da i = a j A a ~ ( i , j , k cyclic). The matrix /3(t) is diagonal and parametrized by

/3 = diag(fl+ + v ' 3 f l _ , f l + - v ' 3 f l _ , - 2 f l + ) . (6)

Rk~ is a general three-dimensional rotation matrix parametrized by the three t ime-dependent Euler an- gles O, ~, g. The superspace metric dS 2 = G ; y dg# ×dgk~ of (3) is in this case reduced to [13]

d S 2 = 24Cabra6 b, (7)

where a and b run from 0 to 5, C.b is diagonal,

Cab = exp(6a) d i a g ( - 1, 1, 1,½ sinh 2 ( 2 v ~ f l _ ) ,

½sinh2(3,8+ + v~,8_) , ~sinh2(3fl+ - x/3fl_)),

(8)

and the set ko of basis one-forms in superspace are defined by

gr a = ~rav dq v

= (da, dfl+, dfl_, sin g dO - cos gt sin 0 d(o,

cos ~/dO + sin~u sin Od~,d~u + cosOdfo), (9)

where dq" = (d~,dfl+, dfl_,dO, d~0,d~). The su- perspace is the direct product of a euclidean space (a) and the five-dimensional symmetric space SL (3) / SO(3) with three linearly independent Killing vec- tors G (s = 1,2, 3) corresponding to SO(3) rotations [13,14].

The spatial part of the integrations in eq. (2) can now be carried out using

f d3x X / ~ = (47Z) 2 exp(3a).

The constraints ~ zt ;g = 0 reduce to G"(q)P~ = O, s = 1, 2, 3. Hence, the new first order action including all constraints reads S = f dt L ( t ) ,

- - - - " ~ r S V

L ( t ) = p.[l ~ - N ( t ) H - N ~, p . , (10)

with new Lagrange parameters N, . ~ and

H = ~ [G"Up.pu + e x p ( - 2 ~ ) (V - 1) + 2A--I,

= A / 9 n 2. (11)

The canonical momenta p , may be expressed lin- early in the n O, but it is, in fact, more instructive to see their relation with the qU which follows from (10) upon variation with respect to Pu-

The metric tensor G~, of the reduced superspace is most conveniently represented by a vielbein e% (t) satisfying e%e~ l, = G~,, ea,,eb" = r/~0 = d i ag ( -1 , 1, 1, 1, 1, 1). Explicitly it follows from eq. (7) that e% = ~ 3"% where some irrelevant prefactors have

been absorbed in A, N, Ns. The potential V in eq. ( 11 ) has the form [3] V = ½tr[exp(4/3) - 2 e x p ( - 2 f l ) + 1 ]. We note that it may be written as

e x p ( - 2 a ) ( V - 1) = G u" O ~ OdP Oqu Oq" '

(12)

with • = -~ e x p ( 2 a ) T r [ e x p ( 2 f l ) ] , generalizing the relation discovered independently in refs. [ 10,15 ]; see also ref. [ 16 ] for a case of non-diagonal/3, including (classical) matter. (Note that due to slightly differ- ing conventions the superspace metric here and in ref. [ 10] differs by a conformal factor, which is irrelevant [6 ] ). There exists therefore the possibility to enlarge the symmetry group under which the action [10] is invariant by adding appropriate "fermion" terms con- taining Grassmann variables gt a.

Let us during a first step eliminate the Lagrange mul- tiplier N by choosing dr = N dt. Then the enlarged action we are looking for is that of the supersymmetric sigma model with metric G~u and superpotential ~ . We therefore have [ 17 ]

dq ~ . ( D ~ D~2 ) L ( z ) = ~ , ~ + ½1 ~u~ Dr Dr ~

- n - 7VSG"p.,= N

(13)

where

D ~ _ O~ua dq" O)uabl[ib, Dr Or dr

(o ) FU) eb a O.)l~,ab ~ --fD~,b a -~- --ea,u - ~ e b + (14)

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Volume 277, number 4 PHYSICS LETTERS B 12 March 1992

defines the covariant derivative and, with q/" = e~" xq/a,

2 ~ 7~u1~# + O q U

+ ½q,:~u (~,*~q/u - ~uuq/*~)

I ]'4* ltt *v it11 t ~tt*g ~tt). -~-~.~.~- v-~. v. +2R. (15)

The constant 2 is left undetermined for the time being. Variation of the action with respect to dq~/dz and 7t~ for fixed q/a, dq/a/dr establishes the relations

n,, = p~ + iOJvabq/*allr tb

}g ~u . (16)

From the canonical brackets (i[ , ] is the Dirac bracket) generalized to include Grassmann variables [181

[q",P~l = i6~, [q/~,q/*~] ~- 6a b (17)

the additional relations

[q~,zrt, ] = iau, [a'~,rgu] = -R~u~q/*~¥ a,

[7~u, q/n] = -i09uabq/b, [Zu, q/a] = -io~uabq/*b (18)

follow. The action (12) is invariant under transfor- mations of the form

6A = i[eQ" + Qe*,A], (19)

where e, e* are Grassmann parameters, and the grass- mannian supercharges Q, Q" satisfy

[Q ,Q ' ] = 2 H , Q2 = 0 = Q.2,

[Q, tI] = 0 = [Q*,H]. (20)

Explicitely they are given by

Q = q/" 7(ea m , + + - ~ .

( oo) (2" = q/'" ½(e~"~,+n,e~ ~ ) - i e ~ ~ . (21)

The constant 2 in eq. (15) is now fixed by (20), but will not be needed in the following. We note that the supercharges are scalars in superspace. Their brackets

with the constraints ~s~p, are proportional to their Lie derivatives along the Killing vectors and there- fore vanish due to the built-in homogeneity under the SO(3) group. E.g., in our coordinates this is trivial to check for {t" = c~ because ~o is cyclic. It follows that the constraints {s~p,, are automatically superinvariam and need not receive any superpartners.

Now we proceed to reinstate time reparametrization invariance and reintroduce d t = d z / N ( t ) , i.e., the conserved hamiltonian is multiplied by N. At the same time we enlarge the global supersymmetry to one local in time by coupling the supercharges to a variable transforming inhomogeneously under supersymmetry transformations [ 19 ]. Thus, we are led to

L ( t ) = ~ " 4 " + ½ i ( 9'aD~'aDt D q /a )D t q/~

- N H - A ~.~ p ~ - ½(Qx + xQ*). (22)

The first ofeqs. (16) and eq. (20) remain unchanged, but the relation between dq"/dt and nu has to be re- calculated and receives additional tcrms containing X and Y*- The action is now invariant under the trans- formation (19) with arbitrarily time-dependent e. (t) if the latter is combined with

6 N ( t ) = Z ' ( t ) e ( t ) + e*( t )Z( t ) ,

6Z(t) = -2i~(t) . (23)

The constraints O = 0 = Q* now foIlow from (22) upon variation with respect to X*,X-

Let us at last also reintroduce the cosmological term and the fermionic panner terms needed to restore su- persymmetry. For this purpose an additional complex Grassmann field q/6 has to be introduced [19], with bracket [ q/6, q/i] = I. Thc cosmological part of the lagrangian then takes the form

( Lcos = - N ( t ) A + ½1 ~ dt

- ½ i V / ~ (¥6X*- Xq/g'),

dt ///6

(24)

which is invariant under the transformation (23) combined with

(25)

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Volume 277, number 4 PHYSICS LETTERS B 12 March 1992

The introduction of the third term on the right-hand

side of (24) thus amounts to the replacements

Q --f Qn = Q + i,fi,

Q” --+ Q: = Q* - iv;&?. (26)

In the final step we now quantize the system replac-

ing the brackets by commutators or anticommutators.

The momenta py then becomep,, = -iG-‘14 (il/aq”) xGIi4 where G = IDet C;,,; and the w”, y/*” a set of

fermion creation and annihilation operators C”, C”’

in a Fock state representation with the anticommuta-

tor [C”, Cb+ ] = qah. The supercharges Qn, Q,’ can then be cast into the form

QA = C”eo” -i 3 + i(L)&Ch+Cc + i a0

aq” as"

+ifiCg,

‘ Q,' = Ca+eo” -i& + iti,,bcCb+ Cc - i g U

-iIL!SC,‘. (27)

The conditions Q? = 0 = Qt’ arc satisfied by (27). All constraints are then satisfied by demanding for

the state vector lul)

Q,,lY', = 0, Q;lW = 0, t.sv~,/lY) = 0. (28)

These first order equations replace the Wheeler- Dewitt equation in the present formulation and gener-

alize those given in ref. [ lo]. Let us remark that these equations are invariant under the conformal trans- formation ea” + Q(qkk”,;? - .Q*(q)J, (q)Y) - 52’/* (q ) (q( Y). Therefore, they automatically satisfy Misner’s [6] criterion for a consistent quantization, which is based on the fact that the conformal factor of the supermetric depends on the coordinate choice and must not influence the quantized description. The operator ordering problem inherent in the Whecler- Dewitt equation therefore does not appear in the present description.

The present paper is not the place to discuss in detail the solutions of eq. (28). However, in order to illustrate in the simplest possible case the influence of the cosmological term and to draw comparisons with earlier work we now drastically reduce the description

396

to the isotropic Bianchi type IX case (i.e., the closed

Friedmann universe) but keep the cosmological term [20,21]. In this case

Q,I = exp(-3a)[-i(a/&) + iexp(2*)] C,

+I UC& P

Q,’ = exp(-3cu) [-i(a/atr) - icxp(2a)] C,,’

-ifiC,+.

The wave function may be written as iY) = (YO +

Y”C,’ +YbC,i + ‘v,,C,’ Cl )iO) and cq. (28) reduce to Yo = 0 = !& and

( -- lti + exp(2cr)) vl, - fi exp(3a) Yh = 0,

( :a + cxp(2u)) Yb - & exp(3cr) Y0 = 0. - (29)

(WC note that for 2 = 0 the function Yb, in general, reduces to the function A+ of ref. [IO].)

For a + -x, i.e., near the initial curvature sin- gularity the cosmological term becomes negligible

and we have [lo] lu, + const. x exp(-@): @ =

{ exp(2a), Ye - 0. The second possibility VI:,, Yb + exp ( + 0 ) can be seen to be excluded when the aniso- tropy variables /3+, p_ are not artificially fixed at zero

[lo], requiring that the wave function dots not di- verge for I/&j + oc, for fixed N. Thereby, the initial bc-

havior of the wave function is uniquely fixed. We note

that the initial state selected is not the Ha&-Hawking state [20], confirming and generalizing a conclusion of ref. [ IO] for/i = 0. On the other hand, apart from the appearance of additional Grassmann variables, it agrees with the initial state found by Vilenkin [21].

A WKB type solution of eq. (29) connected to the

initial condition is

Yh = (a)

const. x { [ 1 - 2;i exp(2a) ]-“‘(t) 1}li2

emi2nl

(30)

0

again consistent with ref. [21]. For exp(2o) > (2z)-’ the exponent is oscillatory and we can select the outgoing wave behaving as

Volume 277, number 4 PHYSICS LETTERS B 12 March 1992

1 ~ 6 ~ (i)const. × exp [ - -~iV'2A exp(3,~)

(,~)

+ ½ i e x p ( c ~ ) / V / ~ - (6A) - l ] (31)

for 2A exp(2cx) >> 1. We conclude that the supersymmetric extension

of quantum cosmology not only gives a definite de- scription of the initial state of the universe [10], but it also describes the subsequent cosmological evolution via the generation of an outgoing wave under the influence of a non-vanishing cosmolog- ical constant (or in an inflationary scenario, of a scalar field, slowly varying with respect to ~). The emerging universe is non-rotating (due to the con- straints ~s"p~]~) = 0) and highly isotropic (due to the behavior ~ ,-~ e x p ( - q ~ ) for c~ << - 1 , and the fact that an outgoing wave emerges on the space like supcrspace surface 2Aexp(6cO = (0q~ /0a ) 2 =

e x p ( 4 a ) [ T r e x p ( 2 / 3 ) ]z). Finally, we jusl mention that a derivat ion of our lagrangian (22) from super- gravity can be given. This derivat ion and a more de- tailed investigation of the solutions of eq. (28) will be presented elsewhere.

This work was supported by the Deutsche For- schungsgemeinschaft through the Sonderforschungs- bereich 237 "Unordnung und groBe Fluktuationen".

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