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With what topology could the universe be created?V G Gurzadyan and A A KocharyanPhysics Institute Ereoan(Submitted 20 July 1988)Zh. Eksp. Teor. Fiz. 95,3-12 (January 1989)The topological and geometrical properties of a self-created universe with cosmological constantare considered in the framework of Hawking's quantum cosmology. The probabilities for creationof the universe with different topologies (including a torus, sphere, hyperbolic space) arecalculated; for an inflationary universe these topologies are found to be equally probable. Theprobability of a quantum change of the topology during the evolution of the universe is calculatedfor a concrete model.

1 INTRODUCTIONThe possibility of a nontrivial topology of the universe

became particularly acute at the beginning of the sixties afterWheeler's work on geometrodynmics, including the idea ofa foamliks:structure of space-time. The recent developmentof ideas about the quantum creation of the universe (see, forexample, Refs. 2-4) have put this question into a somewhatdifferent form, namely, that of the topology with which theuniverse can be created, and with what probability.5 How-ever, as was already noted by Zel'dovich and Starobinski;,difficulties arise already in the very formulation of this prob-lem, in particular, it is not clear what is the meaning of theprobability of creation of a closed universe and how thisprobability is to be normalized.

The framework of quantum cosmology developed byHawking and his collaborator^^^ appears very promisingfor discussing the topology of a created universe. This ap-proach to the determination and interpretation of the wavefunction of the universe has already been used to considermany very important problems relating to the cosmologicalconstant, initial perturbations, the inflationary stage, theCPT theorem, etc9-13 In the framework of the Euclideanformulation of the path integral given by the authors of thesestudies it has been possible not only to define the concept ofprobability but also to find a natural condition of normaliza-tion of the wave function. Another important advantage(particularly in the context of the present paper and empha-sized by Hartle and Hawking7) is the possibility of directcalculations of 3-geometries with nontrivial topology. InRef. Hartle and Hawking also discussed topological prob-lems of the creation of the universe, including the propertiesof a 4-manifold whose edge is a given topologically trivial 3-manifold. These questions were considered in more detail byM kr t ~hya n , ' ~ . ' ~ho for a number of special cases succeededin obtaining restrictions on the properties of a created uni-verse with matter. In this connection it should be noted thatin accordance with cobordism theory and two closed 3-man-ifolds are cobordant (Rokhlin's theorem), i.e., there do notexist restrictions on the topology of a trivial 3 -m an if ~l d. '~

In this paper, using the framework of the approach ofHawking and his collaborators, we attempt to investigateboth the topological and geometrical properties of a createduniverse. In the semiclassical approximation we estimate theprobabilities for creation of a universe with different topolo-gies in superspace. We shall see that the study of this prob-lem requires the finding, for a given value of the cosmologi-cal constant A, of solutions of the Einstein equations for

spaces without matter,

with homogeneous isotropic metric.Among these solutions there are both compact gravita-

tional instantons ~'%nd complex solutions, which, as weshall show, contribute to the required wave function. Weshall call the latter pseudoinstantons.

Having the necessary solutions, we then calculate theEuclidean gravitational action, which occurs in the path in-tegral for the wave function. The method of steepest descentis used then to calculate the integrals of the wave functionsfor spaces with different topologies and thus determine theirrelative probabilities of creation. For spaces capable of un-dergoing an inflationary stage after creation these topologiesare found to be equally probable.

Taking into account the possibility of a quantumchange of the topology of the universe after creation, wecalculate for a concrete example (with A 0) the probabili-ty amplitudes for some transitions. We find that the transi-tion of a sphere into a torus is an event that is extremelyimprobable compared with the transition of a sphere into asphere, i.e., without change of topology.2 CANO NICAL QUANTUM COSMOLOGY

In accordance with the quantum-geometrodynamicformalism, a certain quantum state of the universe is de-scribed by a wave function $(hU that satisfies the Wheeler-DeWitt equation on superspace, i.e, the infinite-dimensionalspace of all Riemannian metrices hq (for a discussion of theproperties of superspace, see Ref. 19). The square of thewave function determines the probability of creation of theuniverse on the 3-manifold S metric hi (in the absence ofmatter).

Hawking and his collaborators assume that the quan-tum state of the real universe is determined by a wave func-tion of the form

where the intergration is over all Cdimensional compactmanifolds M with Euclidean metric g that induces themetric hUon the boundary dM S

The Euclidean quantum action has the form

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whereg s d e t g*, h=det hu,

and is the trace of the second fundamental form of theembedding ofS n M.

If near S the metric gobcan be represented in the formi d ~ ~ = N ~ d t ~ + h i ~ d ~ ~ d d , (3)

then the second fundamental form u s1 dhijK ---- 2 ~ at

In what follows we shall consider isotropic and homo-geneous closed (compact without boundaries) cosmologicalmodels with A term and without matter. In this case themetric on the 3-manifold S, i.e., for t = const, depends on thesingle parameter a:

where

The curvature for the induced metric jsz0

fork = + 1, whenSis the 3-spheres3or the 3-sphere factor-ized with respect to a discrete group S topology); fork = 0,'' when S is the 3-torus T3= S X S X S or anotherflat space T topology1; and for k = - 1, when S s the 3-hyperbolic space H 3 actorized with respect to a discretegroup (H topology )

The space of the metrics ( 3 - 5 ) determines a minisu-perspace. For the metric (5) the action 2 ) has the form

where

In the integral

for the wave function the integration is over all a ( t ) that takethe value a, on S.

Using these expressions, we can estimate in the semi-classical approximation the probability of creation of a uni-verse with k = 0, * 1. But it is first of all necessary to deter-mine the solutions of the Einstein equations that cancontribute to the wave function.3 ISOTROPIC PSEUDOINSTANTONS

To estimate the wave function tC K (a,), it would appearat the first glance that one can proceed as follows. For givenvalue of A we find compact gravitational instantons withmetricgab in the form (3 ). It is known, for example, that forA >0 an instanton solution with metric 3 isS with radius(A/3)- 'I2 (Ref. 18) . Further, if it is found that the metric(3)-(5) cannot be ascribed to the instanton that is foundthen one could expect that the corresponding wave functionmust be zero in the semiclassical approximation.

In fact this procedure is not always correct, since whenwe are calculating the integral in (7) we must take into ac--count not only real, i.e., Euclidean, but also complex solu-tions, (cf. the calculations of integrals by the method ofsteepest descent, when all complex saddle points are takeninto account). Although the physical meaning of such solu-tions is obscure, they nevertheless contribute to the wavefunction.

The saddle points for (7) are found from the Einsteinequation written down for the metric (31,

subject to the condition that there exist t, and t , such thata(t ,)= O, t,

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for A O,k = + 1 4 WAVEFUNCTlONSFORDlFFERENTTOPOLOGlES1 We now turn to the calculation of the wave functions fora(t) = +-sin{H(t(t))). the solutions given above. We first of all calculate the action

with the boundary conditions

1U=Te*Ht a = - s h H ~ a =.H

When N( t) = 1 and { O) = 0 we have for the sign a (t,) =0, a(&)=ao. ( 10)

in HzH1 h t

1a (t)= sin (Ht), We consider the action for H z > 0. Rewriting ( 6 ) in theH formand this is the Euclidean instanton solution S4 . 1;

But when N(t) = - , { O) = r / 2 we have 1~,[a]=-jdtNa[-($)' - k + t ~ o ' ] (11),,a( t)= -ch(Ht),H and using 8),we obtain

-N = l N =

the de Sitter solution, which does not, however, satisfy the 1 00condition ( 9 ) . 1L [ a l = , j d t ~ a [ - 2 ( ~ ) 2 ] = - j d a a ( + )Very interesting is the solution with 1

r-x.op

It can be seen from ( 10) and 11) that for k = 0 the actionwhere 0< E < 1 and the function N( t) is defined continuously I,[a] takes two values,in the interval4'

from which it follows that the wave function isThis solution actually describes creation of the universefrom nothing, making a transition from the Euclidean qO aO) e - ~ ( ~ ) + e - r ~ ( ~ )cos[? ] .hemisphere to the de Sitter stage of expansion (for Ha, > 1 itin fact makes the main contribution to + a,) ).

It is interesting to note that there are some exotic solu- To calculate (a,) fork O we shall proceed from thetions among the ones that we have obtained. Thus, for representati0n (formor see ['I )N(t) = - 1, { O ) = 0 ( k = 1) we have the same 4 1sphere but with proper time $,(ao)m ~ l zj dp exp (+pal3) @h (PI.

wherei.e., with opposite direction with respect to the coordinate 1 itime. This means that a test particle for N = 1 s an antiparti- P= N > a 'cle for N = 1. In this connection we should recall thefundamental question, recently discussed by HawkingI3 and and C is a contour in the complex plane of p parallel to thePage,21of the arrow of time in cosmology. imaginary coordinate axis and to the right of all singularities

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of the function 4, p) : Finally, we have

We note a difference relating to the wave functionk (a ) and to the wave function in the momentum represen-tation, 4, p). In the first case there is a description of cre-

ation from nothing, and a varies from a = 0 to a = a, fork = * 1, while in the second case the momentum of theuniverse arrives at the value p having being created fromdifferent states : from p = c, for k = and fromp,= ki o fork= 1.From ( 13) we can obtain@r*(P)=j d t a l e x p ( - - ~ t a ~ ) ,

where

and the symbols + correspond to the sign of p,. Then thewave function can be rewritten in the form

For each of the integrals in this expression we obtain, afterintegrating by the method of steepest descent (cf. Ref. 7 ) ,

[ (H2a02+l)' 21OS 3H2 4from which we obtain for + (a,)+ a o e x 3 H 2 ]

For Ha, > 1 we obtain similarly

It is now clear why for A = H >0and k = 1 the proba-bility of creation of a universe with a, such that Ha, > 1 is notzero despite the fact that it is not possible to embed S 3ofradius a, in S 4 with radiusH I . The probability is nonzerobecause there exists a complex solution-a pseudoinstanton.

For Ha,> 1 the wave functions take the form0 (a,)= cos F,

1 HaoQ+ a0) exp =)os +) (15)1 Ha,'(ao)= cos =)cos 7),

from which we find the probability ratios

i.e., in this case the probability of creation of a sphere isgreatest. This inequality is not changed for Ha, ,H 2 > 1.One can also estimate the probability of creation of aninflationary ~niverse.~'s is well one of the neces-sary conditions for inflation is a large value of a massivescalar field: m2 p 2 > . Since during this stage the fieldevolves slowly, e /p 4 H , and, therefore, m2e, plays the roleofH2,wecanobtain from (15) fo r H2 >1i.e., the creation of inflationary universes with the S, T , andH topologies is equally probable.5.PROB BILITY OF CH NGE IN THE TOPOLOGY OFTHEUNIVERSE

Thus, we have determined the probability of quantumcreation of a universe with different topologies from noth-ing, i.e., transition from the state a = to a,. Can we nowdraw unambiguous conclusions about the topology of thepresent universe? This is obviously not possible, since thetopology of the universe could have changed during evolu-tion. The classical theory prohibits transitions with a changeof the topology,22but in the quantum theory there are nosuch limitations.Here, considering the example of a toy model, wefind the probability of a quantum change of topology, forwhich it is necesary to extend the mini-superspace represen-tation used above.We consider the case when H =0. Then in the semi-classical approximation the main contribution to the wavefunction (1) will be made by the Euclidean CtorusS X S X S X S = T 4with metric

whereL is a dimensionless constant greater than unity. Into4 Sov. Phys JETP68 1 ) , January 1989 V. G Gunadyan and A. A. Kochgryan

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this torus we can embed any 3-sphere of radiusR 1 these topologies are equally probable.

In the final part of the paper we have considered a toymodel and determined the probability of transitions with achange of topology, a possibility that is, as is well known,permitted by quantum theory. We have shown that forA = 0 transition of a 3-sphere into a 3-torus in a Cdimen-sional torus is strongly suppressed compared with the transi-tion of a sphere into a sphere (with different radius).

We thank D. V. Anosov, B. DeWitt, A. D. Dolgov, Ya.B. Zel'dovich, A. D. Linde, S. G. Matinyan, R. L.Mkrtchyan, D.N Page, and S.Hawking for valuable discus-sions.

As is well known, the geometry defined on the 3-manifold can besmoothly extended to a 4-manifold (if the condition of paracompact-ness is satisfied).

i d - .For k = 0 we require fulfillment of the condition h 'IzHere noncompact solutions are also given.

4 One can show that such functions exist.Without allowance for quantum creation of matter.J . A. Wheeler, in Relativity, Groups and Topology,edited by C. DeWittand B. DeWitt (Gordon and Breach, New York, 1964).

*L.P. Grishchuk and Ya. B. Zeldovich, Quantum Structure ofSpace andTime (Cambridge University Press, 1982).A. Vilenkin, Nucl. Phys. B252 141 (1985).4A.D. Linde, Zh. Eksp. Teor. Fiz. 87, 369 1984) [Sov. JETP 60, 21 1(1984)l.'Ya. B. Zel'dovich and A. A. Starobinskii, Pis'ma Astron. Zh. 10 323

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Translated by Julian B. Barbour

V G Gurzadyan and A A Kocharyan