Is false vacuum bubble nucleation possible in our universe? 
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*Is it possible that a false vacuum bubble to nucleate into our universe rather than a true vacuum one? 

*If yes, it will expand at speed of light within our spacetime or what?
 A: *

*If your theory predicts that we live in a false vacuum, and there exist lower energy states (e.g. true vacuum), then it is possible that a region of space will nucleate a bubble of true vacuum.


Edit: I missed what you were asking. As long as the vacuum that we are tunnelling to is lower, then yes, it should be possible to tunnel to the lower energy state. 
As for bubble nucleation and expansion, there are a few things to be aware of:
2a.) There are sub-critical bubbles and critical bubbles. A bubble that is nucleated will have a outward-pushing (volume) term and an inward pushing (surface-tension) term. Remember that area increases as $R^2$, and volume increases as $R^3$. There is a critical radius where the outward pushing volume term exactly cancels the inward pushing surface term. If you spontaneously nucleate a sub-critical bubble (e.g. $R_{bubble}<R_{crit}$), the surface tension term wins and the bubble will collapse (e.g. the universe outside the bubble will remain in the false vacuum state). On the other hand, if radius of the nucleated bubble is larger than the critical radius, then the bubble will be able to expand due to the volume term overcoming the surface tension term.
2b.) For the most part, the bubble wall moves outward at the speed of light (although I imagine that it might be possible to invent a theory where this is not the case, but I can't think of one offhand). If the bubble is nucleated in a fluid (e.g. when the very early universe was radiation dominated) however, the radiation can act as a viscosity term slowing down the bubble wall. (Although as the radiation dilutes away with the universe's expansion, the bubble wall will eventually expand at the speed of light.)
2c.) If we do presently live in a false vacuum, the future of the universe will depend on the bubble nucleation rate. If the bubble nucleation rate is large enough, then eventually every location in the universe will transition to true vacuum. If the bubble nucleation rate is very small (compared to the expansion of the universe), then there will be a few patches of true vacuum here and there, but the expansion will ensure that the majority of space is in the false vacuum state. In both these scenarios, the every location in the universe will eventually end up in true vacuum (as $t\rightarrow\infty$), but the $\%$ of the overall volume of the universe in true vacuum depends on the nucleation rate.
A: If there is another metastable point below the energy our universe lives in but above the true vacuum level then yes (a nice description on page 3 here). The information within it would travel at the speed of light and any photons that may be within but I am unsure if it might experience an inflationary period or not.
