Scalar field cosmology: I. Asymptotic freedom and the initial-value problem (Q2911100)

Scalar fields are used to fill many gaps in the framework of cosmological theories. In most cases, these fields are treated classically, i. e., with fixed given potentials. However, in the Planck era immediately following the cosmological singularity one expects that one should consider quantum fields, in particular, quantized scalar fields. Since there exist virtual processes with momenta extending to infinity and causing divergences at the high end of the spectrum, the scalar potential is subject to renormalization that cuts off the spectrum and changes with the energy scale. The present paper studies implications of renormalized quantum scalar fields in the Planck era. To this end, Einstein's equations coupled scalar fields are considered for the case of a Friedmann-Robertson-Walker metric. To render the problem tractable it is assumed that the scale parameter (the `radius' of the universe) is identified with the inverse cut-off momentum. This implies that the self-interaction potential should be asymptotically free, a condition that is satisfied by the Halpern-Huang scalar potential derived by renormalization-group analysis. The cosmological equations obtained with this potential show that a vanishing scale parameter is ruled out such that this model starts at some time after the singularity, but still in the Planck era. For a real one-component scalar field, numerical solutions of the equations show that, for large times, after averaging over small rapid oscillations, the Hubble parameter behaves like a power function of time with an exponent between zero and minus one, and the universe expands at an accelerated (exponential) rate. This gives `dark energy' (with an equivalent cosmological constant) that decays such that there does not arise a fine-tuning problem. Under the assumption that the power law for the Hubble parameter will persist, the model is compared with present observations. Finally, the authors address some open issues in their investigation.