Could a pocket universe be birthed out of the negative pressure in our universe? I have just recently started learning about inflationary cosmology so I don't know much about it. But from my understanding, the theory says the big bang was started by the expansion of a small region by negative pressure in the backbone universe that all the other pocket universes expand in. 
If this understanding is correct (please correct me if I'm wrong) then could a pocket universe randomly be birthed out of negative pressure in our universe to give us a pocket universe, within our universe?
If this can't happen why? There something different in the composition of our universe and that of the pocket universe backbone?
 A: I don't know if negative pressure (but see my added edit below) , more importantly there is a theory of inflation, and some good evidence for it. It was caused by a yet unknown inflation field, with its parameters somewhat matching what the cosmic microwave background (CMB) measurements show. [edit added: The field is a quantum field that rolled from a high energy state to a lower equilibrium one, and released a lot of energy. The equivalent equation of state for the energy momentum tensor for it does have the positive energy density and a negative pressure]
As for eternal inflation, it is speculation that if it could happen once it could happen multiple times in different places. Nobody knows and there is no evidence, pure speculation. 
Cosmology matches well the CMB measurements. Including the inflation parameters. Look it up in Wikipedia or many online lectures. Sean Carroll is pure speculation to sell books. 
A: Pocket universes have arisen in different  theories.  Just to name you two, one is alan Gut's inflationary theory idea that Eternal inflation produces pocket universes with all physically allowed vacua and histories. Another is that from sean carroll, who claims that inside every black hole there is an entirely new universe.
