Qualitative study of multidimensional cosmological models

PhysicsLetters B 296 (1992) 307-310 North-Holland

PHYSICS LETTERS B

Qualitative study of multidimensional cosmological models J.C. Fabris Departamento de Fisica e Quimica, Universidade Federal do Espirito Santo, Vit6ria CEP 29000, Espirito Santo, Brazil

and J. Tossa t Laboratoire de Physique Th~orique, Gravitation et Cosmologie Relativistes 2, Universitd Paris- VI, Tour 22/12, 4e ~tage, Bofte 142, 4, Place Jussieu, F- 75252 Paris Cedex 05, France

Received 28 August 1992

A qualitative analysis of cosmologicalmodels induced by the multidimensional theories is performed. The Einstein-MaxweU systemwritten in an arbitrary n-dimensionalmanifold,with n > 4 is studied. Firstly, the original systemis reducedto fourdimensions. The analysis of the resulting equations is made by the dynamical system method. The evolution of the corresponding solutions is represented by curves in the phase diagram. Alsoa comparisonof the results is made with those found previouslyin the context of the couplingof the Einstein-Hilbert iagrangianwith a conformalfield in higherdimensions.

1. Introduction The coupling of a Maxwell field in higher dimensions to the Einstein-Hilbert lagrangian appears in many multidimensional theories. This multidimensional Maxwell field is, of course, no longer conformally invariant, since this property is satisfied exclusively when n = 4. However, when we are constructing cosmological models, such coupling presents some interesting features since, after dimensional reduction, it leads to the Einstein equations coupled to two scalar fields, one coming from the metric in higher dimensions, the other from the "electromagnetic" potential. It is interesting to verify the influence of these scalar fields on the evolution of the Universe. In fact, many speculations about their role during the inflationary phase has been made recently. Here we will study cosmological models constructed from a higher dimensional Einstein-MaxPermanent address: Institut de Math~matiques et de Sciences Physiques (IMSP), Universit6 Nationale du B~nin, B.P. n°613, Porto-Novo,B~nin. URA No. 769, CNRS.

well system in a qualitative way. Firstly, we reduce the original system to four dimensions. We then impose a Robertson-Walker form for the four-dimensional metric. Obtaining a first integral for one of the scalar fields, we reduce the equations to a system of two coupled equations for the scale factors of the external and internal spaces. We study the resulting equations by the dynamical system method. We can represent the possible solutions by curves on a phase diagram, compactified to a circle, the compactification being made with the help of the projection of the whole plane on the Poincar6 sphere. The different kinds of solutions are separated by invariant rays, which correspond to particular solutions, whose expressions can be given explicitly. We find a large variety of solutions. We note first that the dimensional reduction mechanism leads to a variable gravitational coupling. Some curves represent the traditional Friedmann universe with variable gravitational constant G, with a Big-Bang type starting point and a decreasing gravitational coupling, or even an expanding universe with an increasing G. Others represent a collapsing universe, with an

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307

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PHYSICSLETTERSB

increasing or decreasing gravitational coupling. We compare the obtained results with those found in ref. [ 1 ]. There, we have studied in a similar way the multidimensional coupling of the Einstein-Hilbert lagrangian with a conformal type field. In that case, however, we were restricted to even dimensions, due to the conformal nature of the maxwellian type field. This does not happen here. However, the two cases have great similarities: the topology of the solutions does not seem to be affected by the conformal nature of the maxwellian type field.

17 December 1992

where u = • 0. This effective lagrangian characterizes the coupling of gravity with two scalar fields, and leads to the lagrangian of the Brans-Dicke theory, with the parameter o ) = ( 1 - D ) / D and 4 ° = 0 . The variational principle yields the field equations D-1 R,,. - ½g,,.R = - --ffffru~ ( u,,,u,. - ½g,..u,.u .°)

+ u --I ( I~.,,;.-g~.u,~ ;a ) + u -2/0( ~,u ~,, - ½gu. ~,,o ~ ' ° ) ,

2. Description of the geometry and the field equations

u o;°+ ~ 2

The lagrangian density of the theory is defined by ..~=w/~ ( R + FAnF aa) ,

(2.1)

where it is assumed that all the quantities are defined on a (4 + D)-dimensional manifold, We restrict our attention to the metrics of the form (:Is2= gu.dxUdx " - ~ /~,v=0,1,2,3

'°

u --2/D+l 4o ~' a = 0 ,

; o _ ( 2 _ 1~ 4o ~'° = 0 ~D ] u "

(2.7a) (2.7b) (2.7c)

If we choose the metric of the external four-manifold to be the Robertson-Walker spatially flat metric g u ~ d x u d x " = d t 2 - a 2 ( t ) (dx2 + dy2+dz 2) ,

26 a b d X a ~ x b ,

a , b = l , 2 .... , D .

(2.2)

the previous field equations read

The non-vanishing components of the Christoffel connection for the metric (2.2) are:

P~b = ( ~ - I 1~,/~~g,

~ab=(1)fl)"U~ab

(2.3)

(F~a are the Christoffel symbols for the external fourdimensional manifold). The non-vanishing components of the Ricci tensor are: Ru. =Ru~ - D ~

-

l~,u;. ,

/~o = [ q~.o;O+ ( D - 1 )Cb,o~'°]~ab

2D~ - ~,o;*-D(D-

(2.8b) /i+ 3 da ti+ b -2- ~ u -2/o+1 ~ = 0 ,

(2.8c)

~0+ 3 -fia~ + DD----~2~utiu= 0 .

(2.8d)

(2.4) A first integral for eq. (2.8d) is

and the scalar curvature is equal to R=R-

2 ( a ) + ( a ) 2 D - l f t i ~ 2 //_2a_'_fi_u_2m~ t2 ~a/ ~ a : = -TO- ~,u: - u a u T'

(P=ka-3u-(n-2)/D .

1 ) • - 2 ~ , o ~ ,o. (2.5)

Making the only non-null component of the twoform field to he F ~ = ~ , the lagrangian density (2. i ) can be reduced to the effective lagrangian [ 2 ] L=xf-S-g(uR+D-I D 308

u.oU"° u

~o~P'~ U21D-I] '

(2.6)

The term u-2/°~b2 can be removed by the use of (2.8c) in (2.8a) and (2.8b). This yields the system /i u

12 (~)2 D+2

2(D-I)(~)2_3D+6tifi D(D+2) D+2au (2.9a)

PHYSICSLETTERSB

Volume296, number 3,4

D + 2 \a}

a

D - 4 ~i li

° ' (5

Z ( 12X2+ D2+4D-2D y2+

D+2au"

D(D+2)

17 December1992

(2.9b)

+Z(_3DX2+

3(D+6)XY)dX

DD---fly2-(D-4)XY)dY

Setting x =~ / a and y= f~/ u= D dp/ ~, eqs. (2.9) lead to the autonomous dynamical system 3D ~=-b--~x2+

D-I y2 D(D+2)

D ~ -4

=0.

D+2 xy

The singular points at infinity have projective coordinates (X, Y, 0), solutions of

=P(x,y), 12 ~=-b--~x 2-

D2+4D-2y2_ 3r ( Dx + 6y) D(D+2)

=Q(x,y).

12X3+2 ~

D+2

(2.10)

The variable x is the expansion rate of the scale factor of the external space, and the variable y is the expansion rate of the scale factor of the internal space; it can also be interpreted as the measure of the variation of the gravitational coupling G [ 3,4 ]. We wish to study the qualitative evolution of the solutions of this system by the use of the dynamical system techniques approach. This qualitative study will be done by constructing the phase diagram. The phase space is completed by a boundary at infinity (the Poincard' sphere [5,6 ] ).

3. Singular points and invariant rays of the dynamical system

The point (x = 0, y = 0) is a stationary point of the system (2. I 0). It corresponds to the minkowskian space-time. We study the singular points at infinity by using the transformations [5,6 ] X x=--,Z

Y Y=Z'

(3.2)

X2+y2+Z2=l"

Xy2+ 18YX2+ D ~ I y3=0 '

X2+ y2= 1.

(3.3)

Solving this algebraic system, we see that (2.10) has six singular points at infinity A, A', B, B', C, C', which determined three invariant rays AA', BB' and CC' (see figs. 1,2).

B'

Fig. I.Phase diagram of the system in the caseD= I.

B

(3.1)

The equation - Q(x, y) + P(x, y) = 0 deduced from (2.10) yields in terms of the projective coordinates (X, Y, Z)

A' B'

Fig. 2. Phase diagram of the systemin the case D >I2. 309

Volume 296, number 3,4

PHYSICS LETTERS B

A, A' are singular points at infinity in the direction X p=-½, B, B' are in the direction

-x=2,=-½+ Y

N/ 1 2 D '

C, C' in the direction Y

N/ 12D"

The case D = 1, where 21 =0, 22= - 1 is of particular interest. In this case we have a static solution ~ = 0 represented by BB' in fig. I. The topology of the phase diagram is determined by the nature of the singular points. The investigation of this nature leads to the phase diagram shown in figs. l, 2. Analytic expressions can be given for solutions represented by the invariant rays AA', BB' and CC'. For AA', we have a ( t ) ~ t D/tD+2),

17 December 1992

time, because the transformation x - , - x , y - , - y leaves the second member of (2.10 ) unchanged. Solutions represented in parts II and V have trajectories coming from a singularity, but tending to a final state, which is represented by an oversingularity at infinity. These regions are unphysical regions of the phase space. Part VI represents solutions beginning from a singularity, expanding to Minkowski space-time like in part I; the difference between parts I and VI being in the fact that G increases and then Dirac's hypothesis fails for solutions reprcscnted in region VI. Part III is obtained from VI by inversion of time. We observe that this phase diagram is essentially similar to that of ref. [ 1 ]. So, the conformal or nonconformal nature of the "Maxwell" field has no consequence regarding the feature of the solutions. Here however, wc may have even or odd dimensions. We could also point out that, for D = 1 we have a Minkowski space-time with an internal dimension that can increase or decrease with time. This is an unstable property of the system that we have studied.

(I)(t) ~ t -2/(D+2) ,

Acknowledgement

and for BB' and CC'

a ( t ) ~ t ~', ~ ( t ) ~ t p',

3ai+Dfli=l,

with Oli=32i+1,

fli-- 3 2 i + 1 '

i=1,2.

We would like to thank Professor R. Kerner for recommending this analysis and for useful advice. One of us (J.T) is indebted to Professor G. Le Denmat for hospitality at the Laboratoire d'Astronomie et d'Astrophysiquc de Montepellier.

References 4. Results and comments

Solutions lying in thc sector AOC' of the diagrams (part I) are of the "Big-Bang" type: beginning from an original singularity, and expanding to Minkowsky space-time, with G increasing at the beginning and decreasing at the final stage of the evolution, satisfying Dirac's hypothesis [ 3,4 ]. pan IV is obtained from ref. [2] by inversion of

310

[ 1] J. Tossa, J.C. Fabris and C. Romero, C.R. Acad. Sci. Paris 314, Sdr. II (1992) 339. [2] J.C. Fabris, Phys. Lett. B 267 ( 1991) 30. [3] C. Romero, H. Oliveraand J. NeUoNero, Astrophys. Spac. Sci. 158 (1989) 299. [4] C. Will, Theory and experiment in gravitational physics (Cambridge, U.P., Cambridge, 1981 ). [5] G. Sansoni and R. Conti, Non-linear differentialequations (Pergamon, Oxford, 1964 ). [6] O.I. Bogoyavlcnsky, Qualitative theory of dynamical systems in astrophysics and gas dynamics (Springer, Berlin, 1985 ).