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Astron. Astrophys. 353, 1-9 (2000)

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4. Attribution of the vacuum energy to a scalar quantum field

In this section it is shown that the vacuum energy can be attributed to a scalar quantum field [FORMULA] in essentially the same way as by Starkovich & Cooperstock (1992) and by Bayin et al. (1994), only some slight modifications being necessary. For clearness the essential steps of their approach is briefly recapitulated.

The scalar field [FORMULA] that is responsible for the vacuum energy density is conformally coupled to the Ricci-curvature R and is described by a generalized Klein-Gordon equation

[EQUATION]

where [FORMULA] is the scalar field potential and [FORMULA] a numerical constant. According to Birrel & Davies (1982) the energy-momentum-tensor of the field is

[EQUATION]

With this from Einsteins field equations for [FORMULA] the equations

[EQUATION]

[EQUATION]

are obtained where

[EQUATION]

Differentiating (32) with respect to time t yields

[EQUATION]

and eliminating [FORMULA] from this equation with (33) yields [FORMULA] and

[EQUATION]

From (32) and (33) [FORMULA] can be eliminated yielding

[EQUATION]

From this equation the time evolution of S can be determined independently of the evolution of the field [FORMULA] if [FORMULA] is prescribed. Bayin et al. (1994) made the simplifying assumption that during different eras in the evolution of the universe the quantity [FORMULA] defined in (34) assumed different but constant values, especially a very small one, [FORMULA], in the inflationary era.

With slight modifications these ideas can be incorporated into the present model. For this purpose it is assumed, that the field [FORMULA] can be present in addition to primordial matter of equal density [FORMULA] and has the same properties as without matter. (In order to explain the primordial equipartition between energy of the field [FORMULA] and energy of matter, some interaction should be present, but this has to be so small that it can be neglected for the dynamical evolution.)

With this assumption, Eqs. (32)-(33) for [FORMULA] must be replaced by the equations

[EQUATION]

and Eq. (35) must be employed instead of the former assumption [FORMULA] that is recovered from (35) for [FORMULA].

For an equilibrium [FORMULA] and [FORMULA] must be satisfied, and in order to obtain the same equilibrium as in Sect. 2 the assumption [FORMULA] must be made. With this, from (36)-(37) one obtains [FORMULA] and

[EQUATION]

as equilibrium conditions. The dynamics of deviations from equilibrium is obtained from from (32) with (4) and (35), and similarly as (9) one now obtains the equation

[EQUATION]

for it. With [FORMULA] the same results as in Sect. 2 would be obtained. An evolution similar to that obtained by Bayin et al. (1994) can be achieved by assuming that [FORMULA] increases as soon as the equilibrium is left, and saturates at the small value [FORMULA] in order to obtain inflation.

For studying the separation of [FORMULA] from the equilibrium value [FORMULA], the ansatzes

[EQUATION]

and

[EQUATION]

with some constant [FORMULA] are made. Expansion of (39) with respect to [FORMULA] up to terms of order [FORMULA] yields

[EQUATION]

and for real solutions [FORMULA] must be assumed in addition. With these assumptions an expansion evolution of the universe from an initial equilibrium state through instability becomes possible as in Sect. 2.

For [FORMULA] with the assumption [FORMULA] Eq. (39) can be approximated by

[EQUATION]

and yields an evolution

[EQUATION]

Since [FORMULA] is very large, this is an inflation-like evolution although algebraic instead of exponential. At some stage and with similar assumptions and consequences as in Sect. 2 a phase transition described by a rapid change of [FORMULA] must take place in order to enable the transition to a Friedmann-Lemaître evolution.

In the primordial equilibrium state [FORMULA] viz. [FORMULA], [FORMULA] and [FORMULA]. With this from (29)-(31) the condition

[EQUATION]

is obtained, and with the choice [FORMULA] it can be achieved that [FORMULA] has an extremum. [FORMULA] is an arbitrary function and since no condition is obtained for [FORMULA], it appears that this quantity, which is contained in (29)-(31) and (33), can be chosen such that the extremum becomes a minimum.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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