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

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2. The "soft bang" model

The equations for a homogeneous and isotropic universe to be used are the well known Lemaître equations

[EQUATION]

[EQUATION]

([FORMULA] gravity constant, [FORMULA] cosmological constant, [FORMULA] speed of light, [FORMULA], [FORMULA] cosmic scale factor, [FORMULA] mass density and [FORMULA] pressure) following from Einsteins field equations. Eq. (2) is a consequence of Eq. (1) for [FORMULA] when the energy equation

[EQUATION]

is employed.

Concerning the cosmological constant, as in the usual scenarios of cosmic inflation it is assumed that it can be explained and assumes a very large value due to the presence of some primordial vacuum field which at early times and very high temperatures was in a ground state of very high energy density. There are two possible ways to incorporate this assumption into Eqs. (1)-(2):

1. One can set [FORMULA] and replace [FORMULA], where [FORMULA] is the mass density of a quantum field corresponding to its energy density, and [FORMULA] is the mass density of matter plus radiation. In this case the equations must be supplemented with the ansatz [FORMULA] for a negative pressure of the vacuum (Starobinsky 1980 and Zeldovich 1968), and it must be assumed that [FORMULA] adds to the pressure [FORMULA] of radiation and matter, yielding the total pressure [FORMULA].

2. Equivalently one can set [FORMULA], [FORMULA] and replace [FORMULA] by [FORMULA]. If in addition the equation of state for relativistic matter and radiation, [FORMULA], is employed, then from (3) both representations yield

[EQUATION]

where [FORMULA] and [FORMULA] are constant values of [FORMULA] and S to be specified later, and from (1)-(2) both representations yield

[EQUATION]

Now the first step is to look for steady state solutions. For these [FORMULA] is required, and from [FORMULA] and (6) the condition

[EQUATION]

is obtained. It implies that in the static initial state from which the universe is supposed to evolve, there must be an equipartition between the energy of the vacuum and the energy of relativistic matter and/or radiation. Using this result, from the second requirement [FORMULA] and from (6) the conditions [FORMULA] and

[EQUATION]

are obtained. It must, of course, be expected that, like Einstein's static universe, this static solution will be unstable. However, it is just this property which opens the possibility that the present universe has evolved from it through an instability.

The instability of the static solution (8) follows from the existence of a dynamic solution that asymptotically converges towards it as [FORMULA]. To prove this and to derive the unstable solution it suffices to solve Eq. (5) because for [FORMULA] Eq. (6) will then be satisfied automatically. In order to satisfy the condition imposed on the asymptotic behavior, [FORMULA] and (8) must be inserted in Eqs. (4) and (5). However, it is more convenient to use (8) and replace [FORMULA] with

[EQUATION]

and from (5) one thus obtains

[EQUATION]

or

[EQUATION]

For the equation with the plus sign one easily finds the solution

[EQUATION]

with [FORMULA] being an integration constant. (The solution for the minus sign, obtained from this solution by replacing [FORMULA], has [FORMULA] instead of [FORMULA] and is of no interest here because it describes a contracting universe.) For the solution (11) the universe has already existed for an infinite time and was separated from the static solution (8) through an instability in the infinite past. First it starts expanding very slowly, the expansion velocity [FORMULA] becoming larger and larger with time until the exponential term becomes dominant and the expansion exponentially inflating according to

[EQUATION]

just as after a big bang. During inflation the matter is getting extremely diluted and, as a consequence of this, cooled down as well. Therefore, in principle it would be necessary to replace (4b) at a certain stage by the equation [FORMULA] valid for cold matter. However, this is not necessary because at this stage the matter density [FORMULA] is much smaller than the vacuum density [FORMULA] and can thus be neglected.

It must be assumed that through the process of inflation a phase transition is triggered, transforming energy of the vacuum field into energy of matter as in the usual inflation models. From the simplifying assumption that this transition occurs instantaneously at the scale-factor [FORMULA] of a Friedmann-Lemaître solution for regular matter extending until today, and from (11), the condition

[EQUATION]

is obtained, which is satisfied by the choice

[EQUATION]

of the integration constant [FORMULA].

Since in the phase of inflation, according to (4), radiation and relativistic matter of the joint density [FORMULA] are becoming extremely diluted, it must be assumed that energy of the vacuum field with density [FORMULA] is transformed into energy of regular matter from which the present matter of the universe derives. Therefore, the time [FORMULA] must be before the creation of quarks but late enough that no remarkable density of magnetic monopoles was able to develop, which would happen for [FORMULA]. With the choice [FORMULA] corresponding to [FORMULA], according to (8) the intersection of the solution (11) with a Friedmann-Lemaître solution [FORMULA] (obtainable from (12)) occurs at the time [FORMULA] with [FORMULA], and both conditions are met. The radius [FORMULA] from which the universe started according to the present model is well above the Planck length by a factor of [FORMULA]. Thus the present model is far out of the regime where quantum gravity has to be employed. Fig. 1 shows the time evolution of [FORMULA] for the present model ([FORMULA]), for a big bang model with inflation ([FORMULA] and [FORMULA]), for the BP-model ([FORMULA]), and finally for the IR-model ([FORMULA]).

[FIGURE] Fig. 1a and b. Evolution of [FORMULA] for the model of this paper in comparison with the IR-model, the BP-model and a big bang model a for times before and after the Planck-time in linear time scale, and b for [FORMULA] in logarithmic time scale. In the linear time scale according to the IR-model [FORMULA] increases so rapidly that it coincides with the S-axis, while for the big bang model [FORMULA] is still so small that it coincides with the t-axis. For [FORMULA] in logarithmic time scale the present model and the big bounce model become indistinguishable. After inflation all models merge into the same Friedmann-Lemaître evolution.

A question of considerable interest is how and why the phase transition from inflation to ordinary Friedmann-Lemaître expansion is triggered. A dynamical evolution, for example some instability of the vacuum field, must be supposed, and it must be assumed that it was not present in the infinite time before inflation. Qualitatively it may be expected that, as in the inflation scenarios of big bang models, extreme temperatures before the inflation give rise to thermal fluctuations that change the thermally averaged potential of a quantum field in such a way that it has a large positive minimum value. The phase transition can then be attributed to a decrease of this minimum caused by the rapid cooling through inflationary expansion.

During the phase transition, the extremely low values of density and temperature to which the primordial matter and/or radiation have been brought down through inflation must be restored to the high values that are required as initial values for the Friedmann-Lemaître evolution finally leading to the present state of the universe. For this process several possibilities exist:

1. The primordial matter remains as cold and diluted as it came out of the preceding inflation phase and is cooled down and diluted still further during the following Friedmann-Lemaître evolution. In this case the vacuum energy density [FORMULA] must be completely converted into the density of hot matter and radiation, and matter of a kind completely different from the primordial matter that existed for [FORMULA] could be created. As a consequence, in addition to the matter that developed during and after the phase transition there could be a second component of quite different matter, although diluted and cooled down extremely.

2. The vacuum energy could be completely used up for re-heating the primordial matter, at the same time increasing its density due to the mass contained in its thermal energy.

3. The third possibility consists in a combination of re-heating of old matter and creation of new matter from vacuum energy.

In this paper, no preference to any of these possibilities is given because the Friedmann-Lemaître evolution following the phase transition is the same for all of them.

In the following differences between the present model and big bang models are searched. Before the phase transition the present model provides much more time than big bang models, and also the ratio [FORMULA] is quite different, because in the big bang model the density [FORMULA] has already dropped to [FORMULA] when it has reached the scale factor [FORMULA], while in the present model [FORMULA] at this stage. Therefore, the stability behavior with respect to perturbations that locally destroy the high symmetry of the cosmological principle may be quite different. However, it must be expected that differences arising this way are washed out by the exponential inflation. Furthermore, in the process of phase transition, in spite of quite different values of [FORMULA], no differences can be expected to arise, because in both models [FORMULA] is extremely diluted so that it can be neglected in comparison with [FORMULA] which is the same in both models.

After the period of inflation, both models merge into a Friedmann-Lemaître evolution first described by the Lemaître equation

[EQUATION]

and later until today by the Friedmann equation

[EQUATION]

A complete discussion of all possible solutions is given in a survey paper by Felten & Isaacman (1986).

Since during the phase transition most of the vacuum energy is used up for re-heating and/or creation of matter, after it [FORMULA] must be either zero or have an extremely small value in comparison with its value before the phase transition. In order to obtain a solution that complies with the matter density presently observed in the universe, [FORMULA] must be different from zero.

As long as S is sufficiently small, the [FORMULA]- and the k-term on the right hand side of (12) can be neglected and the evolution of the present model and big bang models is essentially identical. Thus the possible appearance of differences is restricted to later times. From the Friedmann equation, for the quantities

[EQUATION]

with

[EQUATION]

the condition

[EQUATION]

with [FORMULA] Hubble parameter can be derived and becomes

[EQUATION]

for the present model ([FORMULA]) or the standard model ([FORMULA] and [FORMULA]) respectively. Furthermore, for the deceleration parameter [FORMULA] the result

[EQUATION]

is obtained.

In the present model, as in the standard model, the choice [FORMULA] is possible, and in this case no difference between the two arises if [FORMULA] is chosen in the latter as well. In the standard model [FORMULA] for [FORMULA], and when in the present model [FORMULA] is zero and [FORMULA] is only slightly above 1, then the differences between the two models are again negligible.

The values [FORMULA] that have to be chosen in the cases [FORMULA] and [FORMULA] considered so far are much larger than the value [FORMULA] obtained from most observations when luminous matter and the dark matter inferred from galaxy motions are added (see e.g. Riess et al. 1998), and they can be only explained by assuming large amounts of as yet unobserved dark matter. In the standard model the observational value [FORMULA] can be only obtained for [FORMULA] while in the present model [FORMULA] is required. With these choices the differences between the two models are remarkable: In the standard model the expansion is decelerated ([FORMULA]) while it is accelerated in the present model ([FORMULA]), and in the present model the scale factor [FORMULA] of today and the life time [FORMULA] of the universe after the phase transition, that can be calculated from (16) and (12), are much larger than in the standard model. For convenience the values of [FORMULA], [FORMULA] and [FORMULA] obtained for the standard model, the present model and other models with [FORMULA] are listed in Table 1 for different choices of the parameters [FORMULA], [FORMULA] and k. For the evaluation of [FORMULA] and [FORMULA] the value [FORMULA] km s- 1 Mpc-1 of the Hubble parameter, having a high probability according to latest measurements (see Riess et al. 1998), was used.


[TABLE]

Table 1. Typical parameter values obtained for the standard model, the present model and other models ([FORMULA] and [FORMULA]) with [FORMULA] km s-1 Mpc- 1. [FORMULA] is given in [FORMULA] light-years and the life time [FORMULA] of the universe after the phase transition is given in [FORMULA] years. (For [FORMULA] no value of [FORMULA] follows from (16).)


The closure of the universe, associated with [FORMULA], could in principle be detected, e.g. by the double observation of a very bright and distant object in opposite directions. So far searches for double observations have not been successful, but if the present model would apply, the result [FORMULA] would explain why.

It can be concluded in summary that pronounced differences between the present model and standard big bang models with inflation only appear if the universe does not contain large amounts of dark matter, and thus [FORMULA] is markedly smaller than 1. The evaluation of observational data some time ago (Liebscher et al. 1992) and just recently (Riess et al. 1998, Perlmutter et al. 1998 and Branch 1998) yields a strong preference for a non-negligible positive cosmological constant of about the magnitude that was employed for the evaluation of the present model in the case [FORMULA]. The recent observational data also confirm the negative value of the deceleration parameter [FORMULA] connected with it. It can be seen from Table 1 that for comparable values of [FORMULA] and [FORMULA] in an open universe ([FORMULA] and [FORMULA]) almost the same results are obtained.

It is illuminating to check the "strong energy condition"

[EQUATION]

that must be satisfied for all times by solutions starting with a big bang singularity (see e.g. Wald 1984). With the assumptions underlying the present model the condition becomes

[EQUATION]

The equilibrium from which the present model evolves through instability has [FORMULA] and thus marks the boundary of the regime in which the strong energy condition is not satisfied; in the later evolution [FORMULA] decreases while [FORMULA] remains constant so the left hand side of the inequality becomes negative and thus recedes from the boundary.

It is possible to attribute the vacuum energy density of the present model to the same kind of scalar field as introduced by Starkovich & Cooperstock (1992) or by Bayin et al. (1994). In this case the treatment is only slightly modified, and the main results are essentially the same as obtained in this section as will be shown in Sect. 4.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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