IL NUOVO CIMENTO VOL. 107 B, N. 9 Settembre 1992

Cosmolog ica l So lut ions of the N = 2, D = 5 Supergravity wi th the Gauss -Bonnet Term.

J. P. DURUISSEAU(1) and J. C. FABRIS (2) (*) (1) Laboratoire de M~canique Relativiste, Universitd Pierre et Marie Curie 4, Place Jussieu, 75252 Paris Cedex 05, France (z) Laboratoire de Physique Thdorique des Particules El~mentaires Universitd Pierre et Marie Curie - 4, Place Jussieu, 75252 Paris Cedex 05, France

(ricevuto il 17 Dicembre 1990; approvato il 13 Marzo 1991)

Summary. - - We study the cosmological solutions of the N = 2, D = 5 supergravity theory modified by the presence of the Gauss-Bonnet term. Our interest is focused on the singularity-free solutions, in both the internal and the external spaces. In fact, there is a family of solutions which possess the desired features.

PACS 04.60 - Quantum theory of gravitation. PACS 98.80 - Cosmology.

1. - Introduct ion.

Studies recently made [1] on the N = 2, D = 5 supergravit9 theory have shown that it is possible to obtain a cosmological compactified solution which has no singularity in the external three-dimensional space. However, this pleasant feature is spoiled by the fact that a singularity in the internal one-dimensional space appears, so that the model comes out to be unstable. In the model we are referr ing to, the effective four-dimensional gravitational coupling is negative before the bouncing, which is a source of instability [2]. Here, we investigate if it is possible to obtain a complete nonsingular solution, in the internal and external spaces, so that gravitational coupling remains always positive. In such a case, the solutions may become stable.

We will t ry to obtain such solutions adding the Gauss-Bonnet te rm to the bosonic sector of that theory. In five dimensions, this te rm gives a nontrivial contribution. However, it is clear that the supersymmetr ic invariance of the original model is broken when we add the Gauss-Bonnet term. Some progress has been made in order to recover the supersymmetr ic invariance; as far as the N = 2, D = 5 supergravity

(*) On leave of absence from Departamento de Fisica e Quimica, Universidade Federal do Espirito Santo - Goiabeiras, Vit6ria, Espirito Santo, Cep 29000, Brazil.

65 - i l N u o v o C i m e n t o B 977

978 ~. P. DURUISSEAU and J. c. FABRIS

theory is concerned, we can quote ref. [3]. On the other hand, it must be said that it is not clear if a complete Lagrangian, and its corresponding transformation laws, may be obtained at all. We will use some partial results obtained up to now.

We will use here the same ansatz as in ref. [1] for the five-dimensional Maxwellian field. This ansatz leads to the desired compactification scheme. We shall also assume that the mean value of the Rarita-Schwinger fields is zero. The presence of the Gauss-Bonnet term leads to highly nonlinear equations. In spite of this, we can obtain some qualitative information about the behaviour of the solutions. Finally, we solve numerically those equations for the case where the solutions seem to present the features we have mentioned above.

2. - T h e f i e l d e q u a t i o n s .

Since there is no Majorana spinor in five dimensions, the N = 2 supergravity is the lowest supersymmetric theory we can construct in this dimension[4]. Its structure is very similar to the N = 1, D = 11 supergravity. If we add the Gauss-Bonnet term, we must profoundly modify the supersymmetric transformation laws and include other terms in the Lagrangian, but it is not clear until now if it is possible to recover completely the supersymmetric invariance, in spite of the fact that some partial results have been already obtained as has been said above. As far as the bosonic sector is concerned, these partial results lead to the following Lagrangian:

(1) L = - (1/4) VgR - ( 1 / 4 ) V g F ~ F A B + (1/6V~)2~'BCDEFABFcDAE +

+ 7 { V g (RABCDRABcD -- 4RABRAB + R2) - (4/S/~) r }"

This Lagrangian represents the interaction of the gravitational field with a five-dimensional ,,Maxwellian, field. The last term may be rewritten as a Lorentzian Chern-Simon term; it is essential to compensate the indesirable contributions coming from the Gauss-Bonnet term under a supersymmetric transformation.

The corresponding field equations are

(2) RAB -- (1/2)gABR + 7{RApQRRB PQR -- 2RApBQR PQ - 2RApRB P + RRAB --

- (1/4) gAB I'QRS (R Rpc~RS - 4RPQRpQ + R2)} = - 2(FApFB p (1/4)gABFpQFPQ),

(8) FARs= (1/2V~)cA~CD~(FBcFDE + yRQpRcRPQDE).

Our conventions are

sgngAB = (+ ), A, B, C... = 1, 2, 3, 4, 5, FAR = ~AAB -- ~BAA �9

We will consider, as in ref. [1], that only the fifth component of the potential AB is not zero. This enables us to write the five-dimensional manifold as a Cartesian product of a four-dimensional manifold and a circle. Considering now that the

COSMOLOGICAL SOLUTIONS OF THE N = 2, D = 5 SUPERGRAVITY ETC. 9 7 9

four-dimensional space-time is maximally symmetric, we have AB = (0, 0, 0, 0, ~") and

(4) ds 2 = dt 2 - a'~dz 2 - (P2dw2,

where dz is the three-space length element, a = a(t) is its corresponding scale factor and ~ = ~(t) is the field representing the scale of the fifth dimension of the metric.

With the metric in the form given by (4), we have for the components of the Ricci tensor

(5) R0o = - 3(~i/a) - (~b/~),

(6) R~j = {a~ + 2a 2 + aa(~/~b)} dij,

(7) R55 = ~b + 3 (a / a )~ ,

and the only nonzero component of FAB is

(8) Fo4 = ~'.

The resulting differential-equations coupling a, ~ and ~" are

(9) 3 ( a / a ) 2 + 3 ( d / a ) ( ( ~ / ~ ) - 6~,(d/a) 3 ( ~ / ~ ) = ((rJ'/~) 2 ,

(10) 2(ii/a) + (~b/r + ( a / a ) 2 + (d /a ) (~ f fP ) -

_ ~ ,{4( i id /a2)( (~/~) + 2 ( d / a ) e ( ~ / O ) } = _ ( ~ / r

(11) 3(d/a) + 3(d/a) e - 6),(iid2/a 3) = (~:/~)2,

(12) ~ + 3(a/a) ~iJ" - (~/r ~}'= 0.

When ~, vanishes, we obtain the same equations of ref. [1], and, when (r"= 0, we find the equations presented in ref.[5] for the five-dimensional case. Here we are interested in the case where k ;~ 0.

The last equation has the solution ~ ' = kq) /a '~, where k is a constant. I t is convenient to reparametrize the time coordinate introducing the variable ~ so that kdr~ = a 3 d t . After defining u = 1 / a and changing y ~ y / k 2, eq. (11) reads

(13) u " + k 2 u / 3 - 2 y ( u " u ' 2 u 4 + u "~u '~) = O.

Equation (9) leads to the following first integral:

(14) u '2 / 2 + k 'Zu 2 - , fu 'au 4 = e ,

where e is the constant of integration. Equation (14) is the equation of a nonlinear harmonic oscillation. The properties of

solutions depend strongly on the sign of ,f and e. When y is negative, e is positive; but, if ), is positive, we have not any a p r i o r i reason to fix the sign of e. I f we rewrite

980 j .P. DURUISSEAU and J. c. FABRIS

1.00

U

0.80

0.60

0.40 7=6.74\ \ \0<7<3

0.20

0.00 2.00 4.00 6.00 ~ 8.00

Fig. 1. - Behaviour of u = 1/a as function of ~ for some values of 7, with the choice of minus sign and 8e = 1.

eq. (14) in integral form, we obtain

(15) f h ( u ) d u = + ( 1 / V ~ ) ~,

where h(u) is given by

(16) h(u) = u 2 { 1 + (Tu 6 _ 8Teu 4 + 1} - 1 .

On the other hand, a suitable combination of equations (9)-(12), leads to the relation

(17) /~ = ~.

We have integrated numerically expression (15) for some values of T, as is shown in fig. 1. Since we are interested in singularity-free solutions--which means that u must remain finite--we have chosen the minus sign inside (16). We observe the existence of a bouncing region for u, as in ref. [1]. But, due to relation (17), the existence of this bouncing always implies a changing of sign for CO: we have consequently a region where the gravitational coupling is negative.

3 . - C o n c l u s i o n .

To obtain a singularity-free cosmological model, it is necessary to turn to the Hawking-Penrose theorem. This can be made, for example, through the inclusion of the exotic type of matter. In general this is made by hand. We tried here to obtain

COSMOLOG1CAL SOLUTIONS OF THE N = 2~ D = 5 SUPERGRAVITY ETC. 981

such feature still in the context of a pure geometric theory. So, we used the Lagrangian of the N = 2, D = 5 supergravity modified by the presence of the Gauss-Bonnet and Lorentzian Chern-Simon terms, which represent the pure geometric terms of an extension of this version of the supergravity theory through the inclusion of nonlinear terms.

The cosmological solutions obtained in the context of the original N = 2, D = 5 supergravity theory reveal the absence of singularity for the three-dimensional scale factor (in a finite cosmological time), but also show a variable gravitational coupling, which becomes negative before the bounce. This is, in that case, an embarrassing feature, since it leads to the instability of solutions, as had already been observed formerly.

We show here that we may obtain a singularity-free three-dimensional scale factor when we add to the five-dimensional supergravity theory the Gauss-Bonnet invariant and its bosonic counterterm. But, the (variable) gravitational coupling changes sign--so, essentially, we find here the same scenario as in ref. [2].

Our search for a bouncing in the three-dimensional scale factor with a positive gravitational coupling is based in the results obtained by Starobinskii [6], who claims the practical impossibility of having an antigravity phase in the evolution of the Universe. Even if this result does not have the force of a theorem, it has been confirmed by the results of ref. [1]. On the other hand, it has been argued that quantum effects in a multidimensional Universe, which can be expressed in a phenomenological way by a quantized scalar field, may lead to a temper- ature-dependent effective gravitational coupling, which may become negative above the critical temperature. This also leads to a bouncing Universe[7,8]. But the definition of an effective gravitational coupling in ref. [7] is not clear, neither is the behaviour of the model against small perturbations. Therefore, the eventual existence of an antigravity phase remains an open question. But the particular results in ref. [2] show that the presence of this phase may lead to an unstable model.

We would like to thank prof. R. Kerner for his suggestions and for many enlightening discussions. J. C. Fabris thanks CAPES (Brazil) for the financial support.

R E F E R E N C E S

[1] R. BALBINOT, J. C. FABRIS and R. KERNER: Cls Q. Grav., 7, 17 (1990). [2] R. BALBINOT, J. C. FABRIS and R. KERNER: Phys. Rev. D, 42, 1023 (1990). 13] S. FERRARA, P. FR~ and M. PORRATI: Ann. Phys., 175, 112 (1987). [4] E. CREMMER: Superspace and Supergravity, Proceedings of Nullfield Workshop (Cambridge

University Press, Cambridge, 1980); R. D'AuRrA, P. FR~, E. MORINA and T. REGGE: Ann. Phys., 135, 327 (1981); 139, 93 (1982).

[5] D. LORENTZ-PETZOLD: Mod. Phys. Lett. A, 9, 827 (1988). [6] A. A. STAROB1NSKII: Soy. Astron. Lett., 7, 36 (1981). [7] M. YOSHIMURA: Phys. Rev. D, 30, 334 (1984). [8] E. I. GUENDELMANN: Los Alamos preprint, LA-UR-90-2104.