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

1. Introduction

In classical big bang models the universe starts at infinite density and infinite temperature with infinite expansion velocity and expansion rate. No reason can be given why it expands so rapidly, the conditions at the big bang enter the models as unexplained initial conditions. In the case of an open universe these conditions must even prevail throughout the infinite space which the universe occupied already at the very beginning. Since for matter densities above the Planck-density  quantum-gravitational effects become important, the validity of classical big bang models is restricted to densities below and times after the Plank time at which this density is reached. Theories describing a quantum-evolution of the universe have been presented by several authors: de Witt (1967), Wheeler (1968), Hartle & Hawking (1983) and others. In general, singularities cannot be avoided in these models either. However, when it is assumed that at very early times there are no particles but only vacuum fields in the universe, then nonsingular solutions also exist, having a similar time evolution as the model considered in this paper (Starobinsky 1980).

A serious problem of former big bang models is the enormously large value of about times the Planck-length obtained for the cosmic scale factor at the Plank time, spanning a range that is not causally connected but nevertheless must be the source of the fantastically isotropic microwave background radiation observed. This problem has been overcome by incorporating into the big bang models a phase of inflationary expansion (Guth 1981, Linde 1982 and Albrecht & Steinhardt 1982), according to which within an extremely short time a causally connected region with the extension of a Planck-length is inflated to the much larger domain obtained from the classical Friedmann-Lemaître evolution. Exponential expansion of an inflationary phase can be obtained for a cosmic substrate with positive energy density and negative pressure (Gliner 1965), and can be explained by the presence of a scalar Higgs field as assumed in the Grand Unified Theories (Georgi & Glashow 1974), other scalar fields, tensor fields or other mechanisms (see Overduin & Cooperstock 1998 for a list of references). The existence of a field of this kind is equivalent to the existence of a - dynamical - cosmological constant, and in the early stages of the universe, the value of the latter must be extremely high, , if the above mentioned difficulty should be overcome.

Thus, there are strong arguments supporting the existence of a large cosmological constant in the very early universe even in the big bang models, classical or combined quantum/classical, and when speaking about big bang models, it will further be assumed that an inflationary phase due to the presence of a large is included. When this necessity is accepted, then cosmological models become possible that avoid the singularities of classical big bang models and even avoid the necessity to invoke a theory of quantum gravity or rather precursors of such a theory which does not yet exist.

A singularity-free model was suggested by Israelit & Rosen (1989) (called IR-model in the following) according to which the (closed) universe was set into existence as a tiny bubble in a homogeneous and isotropic quantum state with a pre-material vacuum energy density corresponding to , with matter and/or radiation density and with the diameter of a Planck length as an initial condition. In this model it is assumed that, because it is in the quantum regime, this state can prevail for some time. At the moment when it traverses the barrier to classical behavior it experiences an accelerated expansion described by the expanding branch of a de Sitter solution. For the later evolution a standard model solution with , and is assumed. The transition from de Sitter inflation into this evolution is performed by a phase transition (Kirzhnits 1972, Kirzhnits & Linde 1972 and Albrecht & Steinhardt 1982), transforming vacuum energy density into ordinary matter as in the inflation scenarios of big bang models. In order to get a smooth connection to a reasonable later evolution, this transition must occur at the very early time when the density of the standard model solution is . This is still far above the density of that prevails in this model at the temperature corresponding to at which GUTs with simple groups give rise to magnetic monopoles of mass ('t Hooft 1974, Polyakov 1974 and Zeldovich & Novikov 1983). Since inflation is already over at this stage, no possibility exists in this model to dilute the heavy monopoles to such a low concentration that an expansion is possible until the present, and that the detection of magnetic monopoles is most unlikely as expressed by the fact they have not yet been observed.

Blome & Priester (1991) proposed a "big bounce model" (called BP-model in the following) in which the universe is closed, exists for an infinite time and first contracts from an infinite size. After it has passed through a minimal radius larger than it starts re-expanding. In the contraction phase the universe is in a primordial state of a quantum vacuum with finite energy density and negative pressure p. The "big bounce" at the minimal radius is supposed to trigger a phase transition by which ordinary matter is created, the vacuum energy density is reduced and the transition into a Friedmann-Lemaître evolution is achieved as in the inflation scenarios of big bang models. The parameters of the model are adjusted in a way such that the above mentioned monopole condition is satisfied.

The IR-model was further developed by Starkovich & Cooperstock (1992) and by Bayin et al. (1994). The pre-material vacuum energy density of the IR-model is attributed to a scalar field obeying a covariant Klein-Gordon equation that is extended by an additional coupling term to the gravitational field and a potential of the field. Furthermore, an equation of state of the form for the pressure p is assumed, for the entropy density the equation of state is derived, and an adiabatic evolution according to = const is found to comply with the other equations. Three different epochs in the evolution of the universe are assumed for each of which is treated as a constant: first an inflationary era with very small , second a radiation era with , and third a matter era with . The starting and end points of the different eras are determined by critical or limiting values of physical quantities like the Planck density or the Planck temperature, the idea of taking these as limiting values for cosmological models being adopted from Markov (1982). As in the IR-model the (closed) universe starts at the Planck density with , and in addition to the IR-model the condition is imposed as initial condition, thus providing a start without singularity. (The model starts at the minimum radius of a big bounce model and, in principle, could be extended further into the past by coupling it to a contraction phase like in the BP-model.) In the inflationary phase, due to the small value of the density decreases only very slowly whence increases, and the inflation is ended when T reaches the Planck temperature . This way no fine tuning is needed for the adjustment to the following radiation dominated era, and in addition, no re-heating is needed because, differently from usual inflation models, the universe enters the radiation era with the appropriate temperature.

Like big bang models with inflation, the model presented in this paper requires some primordial relativistic matter and/or radiation in addition to a high energy density of some primordial quantum field. It can even avoid singularities that are still present in the BP-model, i.e. infinite extension of the closed universe and infinite velocity of contraction in the far past, , and it also avoids the problem of missing monopole dilution.

© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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