  Despite being at the heart of the theory of the "Big Bang" and cosmic
inflation, the quantum field theory prediction of false vacuum tunneling has
not been tested. To address the exponential complexity of the problem, a
table-top quantum simulator in the form of an engineered Bose-Einstein
condensate (BEC) has been proposed to give dynamical solutions of the quantum
field equations. In this paper, we give a numerical feasibility study of the
BEC quantum simulator under realistic conditions and temperatures, with an
approximate truncated Wigner (tW) phase-space method. We report the observation
of false vacuum tunneling in these simulations, and the formation of multiple
bubble 'universes' with distinct topological properties. The tunneling gives a
transition of the relative phase of coupled Bose fields from a metastable to a
stable 'vacuum'. We include finite temperature effects that would be found in a
laboratory experiment and also analyze the cut-off dependence of modulational
instabilities in Floquet space. Our numerical phase-space model does not use
thin-wall approximations, which are inapplicable to cosmologically interesting
models. It is expected to give the correct quantum treatment including
superpositions and entanglement during dynamics. By analyzing a nonlocal
observable called the topological phase entropy (TPE), our simulations provide
information about phase structure in the true vacuum. We observe a cooperative
effect in which the true vacua bubbles representing distinct universes each
have one or the other of two distinct topologies. The TPE initially increases
with time, reaching a peak as the multiple universes are formed, and then
decreases with time to the phase-ordered vacuum state. This gives a model for
the formation of universes with one of two distinct phases, which is a possible
solution to the problem of particle-antiparticle asymmetry.
