Geodesic family of spherical instantons and cosmic quantum creation

TL;DR

This paper formulates Einstein's equations for spherically symmetric metrics and instantons, analyzing the quantum creation probability of different universe models, and finds that only closed models have a nonzero creation probability and vanishing energy.

Contribution

It provides a canonical form of Einstein's equations for spherically symmetric metrics and instantons, and investigates quantum creation probabilities for various cosmological models.

Findings

Only closed Λ-FRW models have nonzero quantum creation probability.

Closed models have vanishing energy, open and flat models have infinite energy.

The results align with previous theories on universe creation and energy conditions.

Abstract

The Einstein field equations for any spherically symmetric metric and a geodesic perfect fluid source are cast in a canonical simple form, both for Lorentzian metrics and for instantons. Both kinds of metrics are explicitly written for the Lema{\^{\i}}tre-Tolman-Bondi family and for a general $\Lambda$-Friedmann-Lema{\^{\i}}tre-Robertson-Walker universe. In the latter case (including of course the instanton version) we study whether the probability of quantum creation of our Universe vanishes or not. It is found, in accordance with previous results, that only the closed model can have a nonzero probability for quantum creation. To obtain this result, we resort to general assumptions, which are satisfied in the particular creation case considered by Vilenkin. On the other hand, Fomin and Tryon suggested that the energy of a quantically creatable universe should vanish. This is in accordance with the above result in which only the closed $\Lambda$FLRW model is quantically creatable while the open and flat models are not. That is so since it can be seen that this closed model has vanishing energy while the open model and the limiting flat case (suitably perturbed) have both infinite energy.