Volume 104A, n u m b e r 4
27 August 1984
ON THE PROBLEMS OF A VERY EARLY UNIVERSE
M.A. MARKOV and V.F. MUKHANOV Institute for Nuclear Research of the USSR Academy of Sciences, Moscow, USSR Received 14 December 1983 Revised manuscript received 15 May 1984
Very early stages of an expanding universe or later stages o f a collapsing universe are considered. F r i e d m a n n ' s universe is assumed to be filled with a nonlinear scalar field. In its development it passes the region near the classical singularity in a phase close to the de Sitter's one. Certain conditions necessary for providing the existence o f a limiting energy density are taken into account in the formalism.
The assumption of the existence of a limiting density as a fundamental law of nature has been put forward in refs. [1,2]. To realize this assumption, the right-hand side of the Einstein equations for a hydrodynamic substance was modified. It turned out that in this case at the initial moment and at the stage of collapse the Friedmann universe must pass in the region near Planck's dimensions in a state close to the de Sitter's one [3,4]. In ref.  also the possibility has been discussed that the fundamental state of matter is some massive scalar field which describes, for example, "maximons". A concrete form o f the lagrangian for such a field has also been proposed with an account of the ideas o f a limiting density. The aim of the present paper is to present a scenario of a very early universe assuming that the universe is filled not with dust, but with a massive scalar field which depends only on time, ~o= ~0(t), and to compare it with some scenarios proposed at the present time. In the general case the Einstein equation is modified as follows  • -- ~k~0 1 i ;l ~0;I)F(~02 /b 2 ) Rki _ ½~/~R = 3t 2 [(~0;/"k + 15~m2~02*(tp2/b2)] ,
where t = (8nG/3) 1/2 is the Planck length, c = fi = I. The functions F and • should have limitations which would make the scenario under consideration close in its 200
properties to the scenarios in refs. [2,3]. In particular we require that the function F should be equal to zero at ~0= b. This condition is necessary in order that a limiting value for the e n e r g y - m o m e n t u m tensor of the field exists. The action for the scalar field can be written as 1
s = f [tp;l~o;lF(~p2/b2) - m2~2~(tp2/b2)] ~
The equations for the field ~0are obtained by varying the action (2):
F~o;k;k + F'~o b -2 ~o;l~o;l+ m2tp(~ + xP'tp2/b 2) = 0 ,
where the prime stands for the derivative with respect to ~o2/b2 . The field ~0= b is a solution of eq. (3) only if ~ ( 1 ) = - ~P'(1). In this case it corresponds to the "vacuum-like" e n e r g y - m o m e n t u m tensor
rki = ~m 2b2 q~(1)6~ ,
or, which is the same, to the A-term in the Einstein equations. Therefore, at ~o= b an isotropic universe is described by the de Sitter solution. The state of the universe at tp = b is the lowest (ground) state in the sense of refs. [5,6]. Now let us investigate the stability of de Sitter's universe under consideration. Let in the Minkowski space (~o < b) the field ¢ correspond to scalar parti-
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27 August 1984
cles of mass m. Then F(0) = q,(0) = 1. Assuming that the functions F and ~ are regular at the point ~0= b and expanding eq. (3) in a power series of ~ -- b - ~0, we derive the following equation f o r y = ~3/2(~ > 0)
y ; k ; k -- m2byl/3 = O.
Hereb - ¢ ' ~ b ,
~2 = ~ [m2/F,(1)] [,l,"(1) + 2'I"(1)]. Note that eq. (5) can be obtained directly from the action (2) by expanding the latter in a power series of (b - ~p) and substituting ~ = b - y2/3. Then the effective potential has the form V ( y ) = _y4/3. As is seen from what follows, the specific form of this potential (V" ~ _oo as y ~ 0) is responsible for peculiarities of the decay in the de Sitter stage. For a homogeneous perturbation ~ ( t ) ( y = y ( t ) ) , •in an expanding de Sitter universe of zero spatial curvature, eq. (5) is rewritten in the form y" + 3Hy - ~n2by 1/3 = 0 ,
where the dot designates differentiation with respect to time t, H = tmb [~ xI,(1)] 1/2, the scale factor a(t) c~ exp(Ht). Let~n 2 > 0 and m ~ m ( i ~ ' ( 1 ) [ ~ lq/'(1)[ [q/'(1) + 2q/(1)i ~ O(1)). Then eq. (5) is valid almost up to ~ ~ b. This condition holds for a wide class of lagrangians. For example, it is valid provided that F = 1 - ~o2/b 2, ¢ = 1 - ~o2/b 2, H = l l m b . By substitution o f z = 23, eq. (6) is reduced by the first-order equation for z (y): z dz/dy = - 3 H z + ~n2by 1/3 ,
whose phase plane is presented in fig. 1. Let us find the analytical form of the most typical solution which passes through the point ~ = 0 (~0 = b) (curve OAC in fig. 1)in asymptotics. In the region OA one can neglect the first term as compared with the second one in the right-hand side of eq. (7). As a result, we have ~o~Tn2bt 2 ,
t<<. a / 3 H .
For the characteristic Hubble time "~I/H the field ~0changes from b to ~ b ( 1 - ~ 2 / H 2 ) . At t > 4/3H, neglecting the derivative in (7), we obtain the solution in the region AC of the OAC curve, which is valid up to t p " b: ~}~n2bt/H,
The de Sitter stage is over when ~0becomes of the order o f b . From this, using eq. (9), we find the duration of the de Sitter stage in question t s ~ ( H / m ) 2 H -1 ~ (/b) 2 n -1 .
For agreement with the observed properties of our universe it is necessary that the duration of the de Sitter expansion stage should exceed 70 Hubble times . In our case this will happen if the critical field b, from which expansion starts, exceeds several times the Planck field, tb ~ 8 - 9 , 1 . The scalar field quantum mass in this case is equal to that of an elementary black hole with m ~ I"- 1 (10-5 g). If the initial field ~0differs from the critical field no more than by the quantity ~ b m 2 / H 2 ~ b/(lb) 2 and the initial values of the derivatives are not very large, the duration of the de Sitter stage in question does not practically depend on the initial conditions and is determined by the expression (10). From this it follows that the duration of the de Sitter stage and, ac-
,1 A long de Sitter stage can also be obtained in the case when the critical field b is exactly equal to the Ptanck one. Then, however, rather strong limitations should be imposed on the possible lagrangian o f the field ~o. For example, for a scalar field with the action
- m 2 ¢ 2 [ 1 + (1 - 2m2/9m2)[(t~) 4 - ~ (f~)2] 1
The duration o f t h e de Sitter stage is equal to t S ~ (m/ ~ ) H -1 . If ~ < 1 0 - 2 m , t h e n t S > 10 2 H -1.
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cordingly, the parameters of the arising Friedmann universe are given by the constants of the theory for a wide class of initial conditions. At the beginning of the de Sitter stage, the energy-momentum tensor may differ from the critical value T~ (¢ = b) by a considerable quantity ~m2b2/(f'b) 2, and all the same the parameters of the arising world will be almost the i _- b). In this respect same as in the case T~ = T~(¢ the model considered does not resemble the hydrodynamic model , where the small difference between the initial and limiting densities strongly affects the duration of the de Sitter stage. The de Sitter model under consideration has common features with different inflation scenarios [ 8 - 1 3 ] , and at the same time does not concretely resemble any of them. Like in all the models considered [ 8 - 1 3 ] , in order to guarantee a long de Sitter stage it is necessary that at some stage the condition H 2 >> V"(y) be satisfied. In some sense our scenario is, naturally, not an exception. The characteristic features that distinguish different inflation models are the following: (a) the initial spatial configuration o f the field ~p, (b) concrete reasons for the fulfillment of the condition H 2 >> V" and peculiarities of the decay of the de Sitter universe. In our case, like in the standard inflation scenarios [ 8 - 1 2 ] , the duration of the de Sitter expansion is determined by the parameters of the theory. However, the character of the discussed instability differs from that conditioned by the possibility of subbarrier tunnelling  or from that connected with the initial quantum fluctuations of the scalar field [9,10] in the models based on potentials of the scalar field with nonlinearities of the type ~kt04, ~p4 ln(~0/~0). Here only due to classical perturbations even the de Sitter model with ~ = b has a finite lifetime which is practically independent of the magnitude of small initial perturbation. At the same time in the inflation models [ 8 - 1 2 ] the duration o f the de Sitter expansion will depend strongly on the magnitude o f small initial perturbation if we disregard quantum fluctuations of the scalar field or the possibility of subbarrier tunnelling. And it is only the account of the latter effect that unambiguously connects the duration of the de Sitter stage with the parameters o f the theory, making this duration practically independent of the initial conditions for a rather wide class of such parameters. The classical character of the investigated instability of the de Sitter universe makes it almost simi202
27 August 1984
lar to the instability in the chaotic scenario of an expanding universe . However, as distinguished from the chaotic scenario, in our case the field ~0is homogeneous at the initial moment (tp ~ b). Such an initial state of the universe may be the ground state in the sense of ref. . The duration of the de Sitter expansion is determined by the parameters of the theory, but not by the initial conditions, which does not resemble the model  and is conditioned by the existence of a separated point ~ ~ b. A long de Sitter stage can be conditioned either by a large initial field ¢ > ~-1 for rather arbitrary functions q~, like in the chaotic scenario, or by a specific form of the "effective" potential, its "smoothing out" (see footnote I). Both the cases can be realized in the considered model. This is not the main feature of our model. Our main goal is, proceeding from the extension o f the ideas about the limiting density to the case o f a scalar field, to show that one can naturally pass over to an effective potential of a rather specific form (V"(y) ~ co at ¢ - b = y2/3 ~ 0) leading to the de Sitter stage of universe expansion with a specific decay due to which the proposed scenario of universe expansion has a number of features common with different inflation models and at the same time essentially differs from them. This scenario unites some characteristic features of the inflation scenarios [ 8 - 1 3 ] . One can regard a massive scalar field as a kind of primary matter, in other words, assume that at the de Sitter stage matter is born only in the form of elementary particles, scalar field quanta o f maximum masses (maximons). Then only a later decay of maximons gives rise to a whole variety of fields in nature after the de Sitter phase. And on the contrary, at the final stage of collapse, under the condition of a limiting density, matter transforms into a gas of maximons, in which form it then passes over into a vacuum-like state of the de Sitter universe. The consideration shows that in our case for a rather wide class of initial conditions (but not for some exceptional ones) in the course o f contraction the universe also passes through the de Sitter stage. On the other hand, if maximons are identified with elementary black holes (m ~ 10 -5 g), which are naturally produced in any form of matter of a maximum limiting density, then all differences among fields vanish, and there remains only the concept of mass density. It is not excluded, therefore, that the hydro-
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dynamic formulation o f the initial phase of the universe (maybe even at the pressure p = 0) is the most exact of its descriptions. In conclusion we note that the limitations imposed on the magnitude of the field ~o, which may result from the fulfillment of the condition F(1) = 0, do not automatically entail a limitation on the e n e r g y - m o m e n t u m tensor since ~ can be given arbitrarily. Nevertheless, if we restrict ourselves to regular solutions of eq. (3), the existence of limiting values of the energy-momentum tensor invariants will follow automatically from the equations. More complex scalar nonlinear fields, for which limitations on the energy density are fulfilled under rather arbitrary initial conditions, are considered elsewhere.
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References  M.A. Markov, Pis'ma Zh. Eksp. Teor. Fiz. 36 (1982) 214.  M.A. Markov, Phys. Lett. 94A (1983) 427.  M.A. Markov, Problems of perpetually oscillating universe, Preprint P-0286 (1983), Inst. Nuclear Research Acad. Sci. USSR.  E.G. Aman and M.A. Markov, Oscillating universe in the case p ~ 0, Preprint P-0290 (1983) Inst. Nuclear Research Acad. Sci. USSR. [5 ] I.B. Hartle and S.W. Hawking, Wave functions of the universe, preprint (1983).  V.K. Mal'tsev and M.A. Markov, The ground state of the closed universe and the problem of the ordering of the operators, to be published.  A.H. Guth, Phys. Rev. D32 (1981) 347.  A.D. Linde, Phys. Lett. 108B (1982) 389. [91 A. Linde, Phys. Lett. l16B (1982) 335.  A.A. Starobinsky, Phys. Lett. l17B (1982) 175.  S.W. Hawking and I.G. Moss, Phys. Lett. l l 0 B (1982) 35.  A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220.  A.D. Linde, Phys. Lett. 129B (1983) 177.