Resonance and instability in a cosmological model

Vistas in Aawnomy, Vol.37,pp. 645-648,1993 Printedin GreatBritain.Allrightsreserved.

0083-6656/93 $24.00 @ 1993 Persemon Pteil Lid

RESONANCE AND INSTABILITY COSMOLOGICAL MODEL

IN A

S. Tasald,* P. Nardonet and I. Prigogine* $ *International Solvay Institutes for Physics and Chemistry, CP231, Campus Plaine ULB, Bd. du Triomphe, 1050 Brussels, B e l g i u m tService de Physique, Brussels Free University, CP231, Campus Plaine, Bd. du Triomphe, 1050 Brussels, Belgium :~Center for Studies in Statistical Mechani~ and Complex Systems, The University of Texas at Austin, Austin, TX 78712, U.S.A.

INTRODUCTION Irreversible processes play an important role in the recently proposed inflationary scenario (e.g., Linde, 1984), where the transition from a false to the true vacua corresponds to the solidification of the supercooled liquid and thus an irreversible process. Moreover, cosmological evolution itself has irreversible character. Prigogine, G~heniau, Gunzig and Nardone (1988) (see also Gunzig, G~heniau and Prigogine, 1987) proposed a phenomenological model of the evolution of the universe where irreversible processes play an essential role both in the evolution of the universe and in the creation of matter. Their model is based on the semi-classical model (Gunzig and Nardone, 1987 and references therein) of the creation of the universe from the empty flat space-time, where the matter field qJ is treated as a quantum mechanical field and the metric g~v is assumed to be conformally flat and is treated as a classical variable" g ~ -- (~/6)~b2~/~v with ¢ a classical variable and 17~v the Minkowski metric. For homogeneous space-time, ten Einstein equations and the equation of motion of the matter field cast into three equations (Gunzig and Nardone, 1987)" ~m2 2

0,2¢ - v2¢ + --F-~ ¢ = 0, I¢m2 ¢2 ) s¢ =o, ~---V-( ~¢m2 2 2- " -~-7~ + ½[((o,¢)~), + (iV¢l=). + ~ + (¢).j---o,

(la)

(lb)

(1<)

where ¢ - ~ ¢@ is the rescaled matter field and (...), stands for the quantum mech&nical average with an substraction of the instantaneous vacuum fluctuation. Let us remark that (lb) follows from (la) and (lc). As discussed by Gunzig and Nardone (1987), the set of equations (1) admits solutions corresponding to the flat space-time and the de Sitter universe. Moreover, when the mass rn of the matter exceeds the threshold value rata = 1 2 7 r V / ~ , the flat-spacetime solution becomes unstable and the de Sitter solution follows. In other words, the creation of matter and the curving of the space-time take place.

646

S. Tasaki et aL

In this report, we reinvestigate the seml-classical model (1) from the point of view of the complex spectral theory, which has been developed by Prigogine and his collaborators (Petrosky and Prigogine, 1991, 1992 and references therein). This theory reveals that, in a wide class of dynamical systems (large Poincar~ systems), the irreversible processes appear as a result of instabilities associated with resonances, and that the characteristic time scales (e.g, lifetimes) corresponding to the irreversible processes can be extracted from the Liouville-von Neumann operator or Hamiltonian of the underlying dynamics as their complex eigenvalues. We show that the instability of the fiat space-time mentioned above can be explained by the resonances between the positive-energy matter and negative-energy gravity. HAMILTONIAN DESCRIPTION In the semi-classical model, the quantum mechanical averages of the rescaled matter field ~ can be used as dynumical variables and provide a Harmltonian description. The relevant variables Pk, •k and Wk are given by (~bk%bkt) ---~p k 6 ( k + k ' ) ,

( ~ k ~ k ' ) ----£ k 6 ( k + k ' ) ,

(~k%bkl)'at-(~bk~kI) -- W k 6 ( k + k ' ) , (2)

where %bkis the Fourier trar, sform of the matter field %b: ~bk =- f ~ - ~ b ( r ) e ik'r. Their equations of motion are given by, with ~ = k 2 + (,¢m2/6)~ 2, Pk ----'.W k ,

]/~rk = 2~k -- 2~2pk ,

~k -'~ - - ~ 2 W k •

(3)

Then it is easily verified that the set of equations (lb) and (3) can be derived as"Hamilton's equations of motion with the Hamiltonian H:

H =--/ dSk ~ [£k + ff:~Pk - 7-x"~] if:' -- 1 2

(4)

by requiring the following Poisson bracket relations: { Pk, Wk' } ~-~4 Pk 6(k - k ' ) ,

{ pk, Ek, } = 2 Wk 6 ( k - k') ,

{ Wk, ~k' } = 4 ~k ~(k - k t ) ,

(5)

{ ~b,p}=l.

In term of these variables, the flat space-time is expressed as ~b= ~b0 (constant), Wk = 0, Pk = (21r)-Sl/(2~t) and Ek = (2~r)-3&t/2. Since one can always scale out constant factors, we will set 40 = V/6V~. STABILITY OF FLAT SPACE-TIME Action-angle descriution of the matter field We first introduce action-angle variables for the matter: Ck -~

(W~ -- 4pkZk) ,

Jk ---~

~

(Zk "4"¢Q2 Pk --

~)(.~k)3) '

( .o21eWk

~0k ~--"t a n - 1 ~,Zk _ 0)2 p k / ,

(6) where tat - ~ is the energy of the matter in the flat space-time. Then as easily seen, the only non-vanishing Poisson bracket is { ~k, Jk' } = 6(k - k').

Resonance and lnstabilhy

647

We then introduce variables similar to the quantum mechanical creation-annihilation operators: bk -- v ~ e -i~°k and b~ - vrJ~e i*'k, which satisfy the Poisson bracket relation ( bk, bk, } = i6(k - k'). Then the unperturbed Hamiltonian is given by Ho =

dSk 2~k Jk -- ~P =

dSk 2w~ b~ bk -- ~p2 ,

(7a)

and the interaction part by V = H - Ho = V1 + 112 + O(b s) with 2^ ¢

vl~-where

¢ --

¢ - V/~

dSk bk+b~ (2~)3/2 ~k '

V2= ~'~

(2~k) 3

'

describes the deviation of the conformal factor.

Generalized canonical transformation We try to eliminate the interaction in the linearized model (7) through a generalized canonical transformation. As discussed by Cary (1981), canonical transformations can be efficiently carried out through Lie transformations Ts = exp({S, .}), where {-..} stands for the Poisson bracket and S the generator. When the generator S is real, the transformation Ts provides a canonical transformation (Cary, 1981) and is unitary. In the complex spectral theory (Petrosky and Prigogine, 1991), the transformation Ts may become non-unitary as a result of the complex valued generator S. Since the Lie transformations preserve the functional form, we have H(bk, b~,, ¢,p) = T ~ I H ( B k , Bk, ~, P), with Bk ---- Tsbk, Bk =- Tsb~, • = Ts¢ and P =- Ts p the transformed variables. Now we determine the generator S perturbatively assuming the smallness of the coupling constant ~; so that the Hamiltonian in terms of new variables has no interactions between the matter and gravity. By expanding T~q , we obtain up to the second order in V~

H = Ho' + Vl - {S, H~} - {S, Vl} + 5,{1S {S, H~}} + V;' + - .. ,

(S)

where H~, V1' and V~ are given by /-/~ _~ Ho(Bk, j~k, p ) _ _~02 ,

VI' = Vl(Bk, j~k, 0 )

V2, = V2(O) + -~2 y~ 2 .

(9)

In (9), fl(,,, O(V~)) stands for the renormalized frequency of the gravity part and will be determined in a self-consistent way. The generator S is then determined so that the term of order of v ~ vanishes: V~ - {S,H~} = 0 , which leads to

H=/dSk2wk~kBk~m4[ - (-y +

dak ~k~-Bk f d3k ' ~k''~-J~k' 24 J (27r)S/2 wk(-~2":-'7--w~) (2~r)S/2 w,,

¢~) + - y ( 1 - - W

(2~)~ ~ ( ~

-

~)) ¢~ •

(10)

The renormalized frequency ~ of the gravity part is determined self-consistently so that the last term of (10) vanishes: 2 ~rn4 f d3k 1 = ~2 ~ - ~ - - j (~r) 2 s ~,k(4~k a 2 -- ~2)

(11)

648

$. Tasaki et al.

Threshold mass From (10), we see that the evolution of the gravity part becomes unstable if (11) admits a complex solution f~. It is easy to see that the solution ~ of (11) should be either real or purely imaginary and that, if m > m:h - 1 2 1 r V ~ , it becomes purely imaginary, i.e., the system becomes unstable. This result coincides with that by Gunzig and Nardone (1987). The appearance of the threshold mass can be explained qualitatively as follows: We observe that, if the frequency ~ of the gravity part exceeds the lower bound 2m of the continuum, i.e., > 2m, pair excitations of the matter and gravity can take place by keeping the total energy zero (which corresponds to resonances between vacuum and excited states) and instability arises. On the other hand, through perturbational arguments, the frequency ~ can be estimated as f~ ~ ~V~-~. Indeed, the integral in the expression of the perturbation 1/'1 has a dimension of (energy) 1/2 and thus can be estimated as ~ v/'~. Then, VI -'~ ~/'~ms/2 ¢ and thus the energy correction A E due to V' can be estimated as A E ,,~ V~/m ~ (v/~m2)2¢ 2 , which serves as a restoring force ~2/2 ¢2 for ¢, or ~ ,~ ~ - ~ 4 . Therefore, the instability arises if ~rn 2 > 1, which is consistent with the previous result. In other words, when the mass exceeds threshold mass, the dressing causes a large energy shift which is enough to induce pair-excitation instability. Before closing the text, let us remark that 1) when we replace negative gravitational energy -1/2 p2 by the positive one +1/2 p 2 the pair-excitation instability disappears, i.e., the negativity of the gravitational energy is essential for the pair-excitation instability, and 2) the present Hamiltonian method, where the quantum mechanical averages play a role of dynamical variables, can be used for the systematic study of the behavior of other semi-classical models. ACKNOWLEDGEMENT We thank Prof. M. Castagnino, Prof. E. Gunzig, Dr. H. H. Hasegawa, Dr. T. Petrosky and Mr. M. Rozenberg for numerous fruitful discussions and comments. We acknowledge the U.S. Department of Energy, Grant N ° FG05-88ER13897, the Robert A. Welch Foundation, and the European Communities Commission (contract n ° 27155.1/BAS.) for support of this work. This work has been partially supported by the Belgian Government under the contract "Pole d'attraction interuniversitaire'. REFERENCES Cary, J.It. (1981) Lie Transform Perturbation Theory for Hamiltonian Systems, Phys. Rep. 7@ 129. Gunzig, E., G~heniau, J. and Prigogine, I. (1987) Nafure 330, 621. Gunzig, E. and Nardone, P. (1987) Self-consistent cosmology~ inflationary universe, and all that... Fundamendals Cosmic Physics 11, 311. Linde, A.D. (1984) The Inflationary Universe. l~ep. Progr. Phys. 47, 925 Petrosky, T and Prigogine, I. (1991) Alternative Formulation of Classical and Quantum Dynamics for Non-integrable Systems, Physica A 175, 146. Petrosky, T. and Prigogine, I. (1992) Complex Spectral Representations and Quantum Chaos, to appear. Prigogine, I., G4heniau, J., Gunzig, E. and Nardone, P. (1988) Thermodynamics of cosmological matter creation Proc. Natl. Acad. Sci. USA 85, 7428.