Do "bubble universes" in eternal inflation have their own spacetime?

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Suekdccia
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TL;DR Summary
In the context of eternal inflation models, can "bubble universes" be described by their own metric, different from the one at the "background"?
In the context of the model of eternal inflation, if an inflating "pocket universe" disconnects from an the background spacetime, does it mean that the baby universe itself can have its own spacetime?

can they be described by a different spacetime metric than the background?

if the original spacetime was asymptotically flat, can the new "universes" be described by an asymptptically flat spacetime as well? (Ref: https://arxiv.org/abs/0706.1233)
 

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