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

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 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 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

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

With this from Einsteins field equations for the equations

are obtained where

Differentiating (32) with respect to time t yields

and eliminating from this equation with (33) yields and

From (32) and (33) can be eliminated yielding

From this equation the time evolution of S can be determined independently of the evolution of the field if is prescribed. Bayin et al. (1994) made the simplifying assumption that during different eras in the evolution of the universe the quantity defined in (34) assumed different but constant values, especially a very small one, , 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 can be present in addition to primordial matter of equal density and has the same properties as without matter. (In order to explain the primordial equipartition between energy of the field 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 must be replaced by the equations

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

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

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

for it. With 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 increases as soon as the equilibrium is left, and saturates at the small value in order to obtain inflation.

For studying the separation of from the equilibrium value , the ansatzes

and

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

and for real solutions 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 with the assumption Eq. (39) can be approximated by

and yields an evolution

Since 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 must take place in order to enable the transition to a Friedmann-Lemaître evolution.

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

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

© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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